\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Yoneda lemma}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{yoneda_lemma}{}\paragraph*{{Yoneda lemma}}\label{yoneda_lemma}

[[!include Yoneda lemma - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{StatementOfYonedaLemma}{Statement and proof}\dotfill \pageref*{StatementOfYonedaLemma} \linebreak
\noindent\hyperlink{corollaries}{Corollaries}\dotfill \pageref*{corollaries} \linebreak
\noindent\hyperlink{corollary_i_yoneda_embedding}{corollary I: Yoneda embedding}\dotfill \pageref*{corollary_i_yoneda_embedding} \linebreak
\noindent\hyperlink{corollary_ii_uniqueness_of_representing_objects}{corollary II: uniqueness of representing objects}\dotfill \pageref*{corollary_ii_uniqueness_of_representing_objects} \linebreak
\noindent\hyperlink{corollary_iii_universality_of_representing_objects}{corollary III: universality of representing objects}\dotfill \pageref*{corollary_iii_universality_of_representing_objects} \linebreak
\noindent\hyperlink{interpretation}{Interpretation}\dotfill \pageref*{interpretation} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{necessity_of_naturality}{Necessity of naturality}\dotfill \pageref*{necessity_of_naturality} \linebreak
\noindent\hyperlink{the_yoneda_lemma_in_semicategories}{The Yoneda lemma in semicategories}\dotfill \pageref*{the_yoneda_lemma_in_semicategories} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{Yoneda lemma} says that the [[set]] of [[morphisms]] from a [[representable presheaf]] $y(c)$ into an arbitrary [[presheaf]] $X$ is in [[natural bijection]] with the set $X(c)$ assigned by $X$ to the representing [[object]] $c$.

The Yoneda lemma is an elementary but deep and central result in [[category theory]] and in particular in [[sheaf and topos theory]]. It is essential background behind the central concepts of \emph{[[representable functors]]}, \emph{[[universal constructions]]}, and \emph{[[universal elements]]}.

\hypertarget{StatementOfYonedaLemma}{}\subsection*{{Statement and proof}}\label{StatementOfYonedaLemma}

\begin{defn}
\label{FunctorUnderlyingTheYonedaEmbedding}\hypertarget{FunctorUnderlyingTheYonedaEmbedding}{}
\textbf{([[functor]] underlying the [[Yoneda embedding]])}

For $\mathcal{C}$ a [[locally small category]] we write

\begin{displaymath}
[C^{op}, Set] \coloneqq Func(C^{op}, Set)
\end{displaymath}
for the [[functor category]] out of the [[opposite category]] of $\mathcal{C}$ into [[Set]].

This is also called the \emph{[[category of presheaves]]} on $\mathcal{C}$. Other notation used for it includes $Set^{C^{op}}$ or $Hom(C^{op},Set))$.

There is a [[functor]]

\begin{displaymath}
\itexarray{
    C &\overset{y}{\longrightarrow}& [C^op,Set]
    \\
    c &\mapsto& Hom_{\mathcal{C}}(-,c)
  }
\end{displaymath}
(called the \emph{[[Yoneda embedding]]} for reasons explained below) from $\mathcal{C}$ to its [[category of presheaves]], which sends each [[object]] to the [[hom-functor]] into that object, also called the [[representable presheaf|presheaf represented]] by $c$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{([[Yoneda embedding]] is [[adjunct]] of [[hom-functor]])}

The Yoneda embedding functor $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ from Def. \ref{FunctorUnderlyingTheYonedaEmbedding} is equivalently the [[adjunct]] of the [[hom-functor]]

\begin{displaymath}
Hom_{\mathcal{C}}
  \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
  \longrightarrow
  Set
\end{displaymath}
under the [[product category]]/[[functor category]] [[adjoint functor|adjunction]]

\begin{displaymath}
Hom(C^{op} \times C, Set) 
    \stackrel{\simeq}{\to}
  Hom(C, [C^{op}, Set])
\end{displaymath}
in the [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal category]] of categories.

\end{remark}
\begin{prop}
\label{YonedaLemma}\hypertarget{YonedaLemma}{}
\textbf{([[Yoneda lemma]])}

Let $\mathcal{C}$ be a [[locally small category]], with [[category of presheaves]] denoted $[\mathcal{C}^{op},Set]$, according to Def. \ref{FunctorUnderlyingTheYonedaEmbedding}.

For $X \in [\mathcal{C}^{op}, Set]$ any [[presheaf]], there is a canonical [[isomorphism]]

\begin{displaymath}
Hom_{[C^op,Set]}(y(c),X) \;\simeq\; X(c)
\end{displaymath}
between the [[hom-set]] of [[presheaf]] [[homomorphisms]] from the [[representable presheaf]] $y(c)$ to $X$, and the value of $X$ at $c$.

\end{prop}
This is the standard notation used mostly in pure [[category theory]] and [[enriched category theory]]. In other parts of the literature it is customary to denote the presheaf represented by $c$ as $h_c$. In that case the above is often written

\begin{displaymath}
Hom(h_c, X) \simeq X(c)
\end{displaymath}
or

\begin{displaymath}
Nat(h_c, X) \simeq X(c)
\end{displaymath}
to emphasize that the morphisms of presheaves are [[natural transformation]]s of the corresponding functors.

\begin{proof}
The proof is by chasing the element $Id_c \in C(c, c)$ around both legs of a [[naturality square]] for a [[natural transformation]] $\eta: C(-, c) \to X$ (hence a homomorphism of presheaves):

\begin{displaymath}
\itexarray{
     C(c, c) 
       & \stackrel{\eta_c}{\to} & 
     X(c) & & & & Id_c & \mapsto & \eta_c(Id_c) & \stackrel{def}{=} & \xi \\ 
 _\mathllap{C(f, c)} \downarrow & & \downarrow _\mathrlap{X(f)} & & & & \downarrow & & \downarrow _\mathrlap{X(f)} & & \\ 
C(b, c) & \underset{\eta_b}{\to} & X(b) & & & & f & \mapsto & \eta_b(f) & & 
}
\end{displaymath}
What this diagram shows is that the entire transformation $\eta: C(-, c) \to X$ is completely determined from the single value $\xi \coloneqq \eta_c(Id_c) \in X(c)$, because for each object $b$ of $C$, the component $\eta_b: C(b, c) \to X(b)$ must take an element $f \in C(b, c)$ (i.e., a morphism $f: b \to c$) to $X(f)(\xi)$, according to the commutativity of this diagram.

The crucial point is that the naturality condition on any [[natural transformation]] $\eta : C(-,c) \Rightarrow X$ is sufficient to ensure that $\eta$ is already entirely fixed by the value $\eta_c(Id_c) \in X(c)$ of its component $\eta_c : C(c,c) \to X(c)$ on the [[identity morphism]] $Id_c$. And every such value extends to a natural transformation $\eta$.

More in detail, the bijection is established by the map

\begin{displaymath}
[C^{op}, Set](C(-,c),X)
  \stackrel{|_{c}}{\to}
  Set(C(c,c), X(c))
  \stackrel{ev_{Id_c}}{\to}
  X(c)
\end{displaymath}
where the first step is taking the component of a [[natural transformation]] at $c \in C$ and the second step is [[evaluation]] at $Id_c \in C(c,c)$.

The inverse of this map takes $f \in X(c)$ to the natural transformation $\eta^f$ with components

\begin{displaymath}
\eta^f_d := X(-)(f) : C(d,c) \to X(d)
  \,.
\end{displaymath}
\end{proof}
\hypertarget{corollaries}{}\subsection*{{Corollaries}}\label{corollaries}

The Yoneda lemma has the following direct consequences. As the Yoneda lemma itself, these are as easily established as they are useful and important.

\hypertarget{corollary_i_yoneda_embedding}{}\subsubsection*{{corollary I: Yoneda embedding}}\label{corollary_i_yoneda_embedding}

The Yoneda lemma implies that the [[Yoneda embedding]] functor $y \colon C \to [C^op,Set]$ really is an \emph{embedding} in that it is a [[full and faithful functor]], because for $c,d \in C$ it naturally induces the isomorphism of Hom-sets.

\begin{displaymath}
[C^{op},Set](C(-,c),C(-,d)) \simeq (C(-,d))(c) = C(c,d)
\end{displaymath}
\hypertarget{corollary_ii_uniqueness_of_representing_objects}{}\subsubsection*{{corollary II: uniqueness of representing objects}}\label{corollary_ii_uniqueness_of_representing_objects}

Since the [[Yoneda embedding]] is a [[full and faithful functor]], an [[isomorphism]] of [[representable functor|representable presheaves]] $y(c) \simeq y(d)$ must come from an [[isomorphism]] of the representing objects $c \simeq d$:

\begin{displaymath}
y(c) \simeq y(d) \;\; \Leftrightarrow \;\;
  c \simeq d
\end{displaymath}
\hypertarget{corollary_iii_universality_of_representing_objects}{}\subsubsection*{{corollary III: universality of representing objects}}\label{corollary_iii_universality_of_representing_objects}

A [[presheaf|presheaf]] $X \colon C^{op} \to Set$ is [[representable functor|representable]] precisely if the [[comma category|comma category]] $(y,const_X)$ has a [[terminal object]]. If a [[terminal object]] is $(d, g : y(d) \to X) \simeq (d, g \in X(d))$ then $X \simeq y(d)$.

This follows from unwrapping the definition of [[morphisms]] in the [[comma category]] $(y,const_X)$ and applying the Yoneda lemma to find

\begin{displaymath}
(y,const_X)((c,f \in X(c)), (d, g \in X(d)))
  \simeq
  \{
    u \in C(c,d) : X(u)(g) = f
  \}
  \,.
\end{displaymath}
Hence $(y,const_X)((c,f \in X(c), (d, g \in X(d))) \simeq pt$ says precisely that $X(-)(f) \colon C(c,d) \to X(c)$ is a bijection.

\hypertarget{interpretation}{}\subsubsection*{{Interpretation}}\label{interpretation}

For emphasis, here is the interpretation of these three corollaries in words:

\begin{itemize}%
\item \textbf{corollary I} says that the interpretation of presheaves on $C$ as generalized objects probeable by objects $c$ of $C$ is consistent: the probes of $X$ by $c$ are indeed the maps of generalized objects from $c$ into $X$;


\item \textbf{corollary II} says that probes by objects of $C$ are sufficient to distinguish objects of $C$: two objects of $C$ are the same if they have the same probes by other objects of $C$.


\item \textbf{corollary III} characterizes [[representable functor]]s by a [[universal property]] and is hence the bridge between the notion of [[representable functor]] and [[universal construction]]s.



\end{itemize}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

The Yoneda lemma tends to carry over to all important generalizations of the context of [[locally small category|categories]]:

\begin{itemize}%
\item There is an analog of the Yoneda lemma in [[enriched category theory]]. See [[enriched Yoneda lemma]].


\item In the context of [[module]]s (see also [[Day convolution]]) the Yoneda lemma becomes the important statement of [[Yoneda reduction]], which identifies the bimodule $\hom_C(-, -)$ as a unit bimodule.


\item There is a [[Yoneda lemma for bicategories]].


\item There is a [[Yoneda lemma for (∞,1)-categories]].


\item In [[functional programming]], the Yoneda embedding is related to the [[continuation passing style]] transform.


\item Formulation of the lemma in [[dependent type theory]]: \href{http://homotopytypetheory.org/2012/05/02/a-type-theoretical-yoneda-lemma/}{A type theoretical Yoneda lemma} at \href{http://homotopytypetheory.org}{homotopytypetheory.org}



\end{itemize}
\hypertarget{necessity_of_naturality}{}\subsection*{{Necessity of naturality}}\label{necessity_of_naturality}

The assumption of naturality is necessary for the Yoneda lemma to hold. A simple counter-example is given by a category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \mathbb{Z}_{\geq 0}$, the set of integers greater than or equal to $0$, in which $Hom(B,A) = \mathbb{Z}_{\geq 1}$, the set of integers greater than or equal to $1$, and in which composition is addition. Here it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is greater than or equal to $1$, so cannot have an inverse, since $0$ is the identity on $A$ and $B$).

A finite counter-example is given by the category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \{0, 1\}$, in which $Hom(B,A) = \{0, 2\}$, and composition is multiplication modulo 2. Here, again, it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is $0$, so cannot have an inverse, since $1$ is the identity on $A$ and $B$).

\hypertarget{the_yoneda_lemma_in_semicategories}{}\subsection*{{The Yoneda lemma in semicategories}}\label{the_yoneda_lemma_in_semicategories}

An interesting phenomenon arises in the case of [[semicategory|semicategories]] i.e. ``categories'' (possibly) lacking [[identity morphisms]]:

the Yoneda lemma fails in general, since its validity in a semicategory $\mathcal{G}$ implies that $\mathcal{G}$ is in fact already a category because the Yoneda lemma permits to embed $\mathcal{G}$ into $PrSh(\mathcal{G})$ and the latter is always a category, the embedding then implying that $\mathcal{G}$ is itself a category!

But for [[regular semicategories]] $\mathcal{R}$ there is a [[unity of opposites]] in the category of all [[semipresheaves]] on $\mathcal{R}$ between the so called regular presheaves that are [[colimits]] of [[representable presheaf|representables]] and presheaves satisfying the Yoneda lemma, whence \emph{the Yoneda lemma holds dialectically for regular presheaves!}

For some of the details see at \emph{[[regular semicategory]]} and the references therein.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item The Yoneda lemma is the or a central ingredient in various [[reconstruction theorem]]s, such as those of [[Tannaka duality]]. See there for a detailed account.


\item In its incarnations as [[Yoneda reduction]] the Yoneda lemma governs the algebra of [[end]]s and [[coend]]s and hence that of [[bimodule]]s and [[profunctor]]s.


\item The Yoneda lemma is effectively the reason that [[Isbell conjugation]] exists. This is a fundamental duality that relates [[geometry]] and [[algebra]] in large part of mathematics.



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Yoneda reduction]]


\item [[co-Yoneda lemma]]


\item [[Yoneda structure]]


\item [[Brown representability theorem]]


\item [[continuation-passing style]]


\item [[regular semicategory]]



\end{itemize}
$\backslash$linebreak

\hypertarget{References}{}\subsection*{{References}}\label{References}

For general references see any text on [[category theory]], as listed in the references \href{category+theory#References}{there}.

The term \emph{Yoneda lemma} originated in an interview of [[Nobuo Yoneda]] by [[Saunders Mac Lane]] at Paris Gare du Nord:

\begin{itemize}%
\item Yoshiki Kinoshita, \href{http://www.mta.ca/~cat-dist/catlist/1999/yoneda}{posting to catlist in 1996}.

\end{itemize}
In \emph{[[Categories for the Working Mathematician]]} MacLane writes that this happened in 1954.



Reviews and expositions include

\begin{itemize}%
\item [[Saunders MacLane]], \emph{The Yoneda Lemma}, Mathematica Japonicae 47: 156 (1998)


\item [[Tom Leinster]], \emph{[[LeinsterYoneda.ps:file]]}


\item Marie La Palme Reyes, Gonzalo E. Reyes, and Houman Zolfaghari, \emph{Generic figures and their glueings: A constructive approach to functor categories}, Polimetrica sas, 2004 (\href{https://reyes-reyes.com/2004/06/01/generic-figures-and-their-glueings-a-constructive-approach-to-functor-categories/}{author page},\href{https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf}{pdf}).



\end{itemize}
A discussion of the Yoneda lemma from the point of view of [[universal algebra]] is in

\begin{itemize}%
\item [[Vaughan Pratt]], \emph{The Yoneda lemma without category theory: algebra and applications} (\href{http://boole.stanford.edu/pub/yon.pdf}{pdf}).

\end{itemize}
[[!redirects yoneda lemma]] [[!redirects Yoneda Lemma]] [[!redirects Yoneda embedding lemma]] [[!redirects Yoneda imbedding lemma]]



\end{document}