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What is sometimes called the co-Yoneda lemma is a basic fact about presheaves (a basic fact of topos theory): it says that every presheaf is a colimit of representables and more precisely that it is the “colimit over itself of all the representables contained in it”.
One might think of this as related by duality to the Yoneda lemma, hence the name.
Throughout, let
$V$ be a closed symmetric monoidal category (for instance $V =$ Set with its Cartesian product);
$\mathcal{C}$ a small $V$-enriched category.
For $c,d \in Obj(\mathcal{V})$ two objects, we write $\mathcal{C}(c,d) \in V$ for the hom-object.
Recall that the (enriched) Yoneda lemma says that for $F \colon \mathcal{C}^{op} \to V$ a $V$-enriched functor out of the opposite category of $\mathcal{C}$, hence a $V$-valued presheaf on $\mathcal{C}$ and $c \in \mathcal{C}$ an object of $\mathcal{C}$, there is a natural isomorphism in $V$
where on the left we have the hom-object in the enriched functor category between the functor represented by $c$ and the given functor $F$.
Using the expression of these hom-ebjects on the left as ends, this reads
In this form the Yoneda lemma is also referred to as Yoneda reduction.
Under abstract duality an end turns into a coend, so a co-Yoneda lemma ought to be a similarly fundamental expression for $F(c)$ in terms of a coend.
The natural candidate is the statement that:
Every presheaf $F$ is a colimit of representables, in that
hence
where $Y$ denotes the Yoneda embedding. In module-language, using the tensor product of functors, this reads
Yet another way to state this is as a colimit over the comma category $(Y,F)$, for $Y$ the Yoneda embedding:
This statement we call the co-Yoneda lemma.
To show that a presheaf $F \colon \mathcal{C}^{op} \to V$ is canonically presented as a colimit of representables, we exhibit a natural isomorphism
By the definition of the coend, maps
are in natural bijection with families of maps
extranatural in $c$ and natural in $d$. Those are in natural bijection with families of maps
natural in $c$ and extranatural in $d$. These are in natural bijection with families of maps
(natural in $c$), where the isomorphism is by the Yoneda lemma. Thus we have exhibited a natural isomorphism
(natural in $G$). By Yoneda again, this gives
The statement of the co-Yoneda lemma in prop. 1 is a kind of categorification of the following statement in analysis (whence the notation with the integral signs):
For $X$ a topological space, $f \colon X \to\mathbb{R}$ a continuous function and $\delta(-,x_0)$ denoting the Dirac distribution, then
If one follows the Yoneda-lemma argument at the end of the proof of prop. 1, one arrives at the explicit isomorphism
Namely, it corresponds to the family of maps
(extranatural in $c$ and natural in $d$) which in turn corresponds to the natural family
associated with the structure of the functor $F \colon \mathcal{C}^{op} \to V$.
Let $V =$ Set, and recall the definition of the coend as a coequalizer
This says that the coend is the set of equivalence classes of pairs
where two such pairs
are regarded as equivalent if there exists
such that
(Because then the two pairs are the two images of the pair $(f,y)$ under the two morphisms being coequalized.)
But now considering the case that $c = c_0$ and $f = id_{c_0}$, so that $g= \phi$ shows that any pair
is identified, in the coequalizer, with the pair
hence with $\phi^\ast(y)\in F(c_0)$, and that this coequalizing operation is the action
of morphisms on elements of the presheaf by pullback.
In a brief uncommented exercise on MacLane, p. 62
the following statement, which is attributed to Kan, is called the co-Yoneda lemma.
For $D$ a category, Set the category of sets, $K : D \to Set$ a functor, let $(* \darr K)$ be the comma category of elements $x \in K d$, let $\Pi: (* \darr K) \to D$ be the projection $(x \in K d) \mapsto d$ and let for each $a \in D$ the functor $\Delta_a: (* \darr K) \to D$ be the diagonal functor sending everything to the constant value $a$.
The co-Yoneda lemma in the sense of Kan/MacLane is the statement that there is a natural isomorphism of functor categories
Here is an outline of an explicit proof:
A natural transformation $\phi: K \to D(a, -)$ assigns to each element $x \in K c$ an element $\phi_c(x) \in D(a, c)$, i.e., an arrow $\phi_c(x): a \to c$. We define a corresponding transformation $\psi: \Delta_a \to \Pi$ which assigns to each object $(c, x \in K c)$ in $(*\darr K)$ the morphism $\phi_c(x): a \to c = \Pi(c, x)$. It is easy to check that the naturality condition on $\phi$ corresponds to the naturality condition on $\psi$, and that the correspondence is bijective.
Here is a more conceptual proof in terms of comma categories:
Set classifies discrete fibrations, in the sense that a functor $G : D \to Set$ classifies the discrete fibration
and natural transformations $\alpha : G \to F$ correspond to maps of fibrations
i.e. functor which commute on the nose with the projections $\Pi_G$, $\Pi_F$ to the base category $D$).
This applies in particular to $F = hom(a,-)$. Notice the category of elements $El(hom(a,-))$ is the co-slice $(a \downarrow D)$, with its usual projection $\Pi$ to $D$.
However, the comma category $(a \downarrow D)$ is the “lax pullback” (really, the comma object, the discussion at 2-limit) appearing in
and so a fibration map $El(G) \to (a \downarrow D)$ corresponds exactly to a lax square
This yields the co-Yoneda lemma in the sense of MacLane’s exercise.
For enrichment over $V = Top^{\ast/}_{cg}$ (pointd compactly generated topological spaces) the co-Yoneda lemma in the sense of every presheaf is a colimit of representables appears for instance as
The co-Yoneda lemma in the sense of MacLane appears as a brief uncommented exercise on p. 63 of
where it is attributed to Daniel Kan.