The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding

Let $\mathcal{C}$ be locally small. The Yoneda assignment $y$ sends an object $a$ to the representable presheaf

$y(a):=\mathcal{C}(-,a):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$

and sends a morphism $f:a\to b$ to the natural transformation $y(f):y(a)\Rightarrow y(b)$ whose component at $c$ is postcomposition by $f$:

$y(f{)}_c:\mathcal{C}(c,a)⟶\mathcal{C}(c,b),g⟼f\circ g\mathrm{.}$

The functoriality and naturality of these formulas are instances of the hom-bifunctor The hom-assignment $\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is a bifunctor.

When $\mathcal{C}$ is small, Functor category $[\mathcal{C},\mathcal{D}]$ and If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small form the presheaf category and the assignment is the functor

$y:\mathcal{C}⟶[{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}],$

traditionally called the Yoneda embedding. For an arbitrary large locally small $\mathcal{C}$, the same formulas are called the Yoneda assignment; no large-source functor category is silently formed. Full faithfulness is proved in The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective. Under the terminology of Embedding and full embedding of categories, a fully faithful functor is a full embedding only if its object map is also injective.