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For $C$ a locally small category, every object $X$ of $C$ induces a presheaf on $C$: the representable presheaf $h_X$ represented by $X$. This assignment extends to a functor $C \to [C^{op}, Set]$ from $C$ to its category of presheaves. The Yoneda lemma implies that this functor is full and faithful and hence realizes $C$ as a full subcategory inside its category of presheaves.
Recall from the discussion at representable presheaf that the presheaf represented by an object $X$ of $C$ is the functor $h_X :C^{op} \to Set$ whose assignment is illustrated by
which sends each object $U$ to $Hom_C(U,X)$ and each morphism $\alpha:U'\to U$ to the function
Moreover, for $f : X \to Y$ an morphism in $C$, this induces a natural transformation $h_f : h_X \to h_Y$, whose component on $U$ in $X$ is illustrated by
For this to be a natural transformation, we need to have the commuting diagram
but this simply means that it doesn’t matter if we first “comb” the strands back to $U'$ and then comb the strands forward to $Y$, or comb the strands forward to $Y$ first and then comb the strands back to $U'$
which follows from associativity of composition of morphisms in $C$.
The Yoneda embedding for $C$ a locally small category is the functor
from $C$ to the category of presheaves over $C$ which is the image of the hom-functor
under the Hom adjunction
in the closed symmetric monoidal category Cat.
Hence $Y$ sends any object $c \in C$ to the representable presheaf which assigns to any other object $d$ of $C$ the hom-set of morphisms from $d$ into $c$:
It follows from the Yoneda lemma that the functor $Y$ is full and faithful. It is also limit preserving (= continuous functor), but does in general not preserve colimits.
The Yoneda embedding of a small category $S$ into the category of presheaves on $S$ gives a free cocompletion of $S$.
If the Yoneda embedding of a category has a left adjoint, then that category is called a total category .
A category is a total category if its Yoneda embedding has a left adjoint.
singleton injection, the Yoneda embedding for 0-category theory.