Yoneda embedding

Remarks

The components $y(f)_X$ are natural in $X$ because $f \circ (g \circ h) = (f \circ g) \circ h$. By the Yoneda lemma with $F = \Hom(-, B)$, the map $f \mapsto y(f)$ is a bijection $\Hom(A, B) \to \mathrm{Nat}(y(A), y(B))$, so $y$ is full and faithful. Since $y$ is fully faithful, $y(A) \cong y(B)$ implies $A \cong B$. The dual functor $A \mapsto \Hom(A, -)$ embeds $\mathcal{C}^{\mathrm{op}}$ into $[\mathcal{C}, \mathbf{Set}]$.

Example: let $P$ be a preorder seen as a category, with one morphism $a \to b$ when $a \le b$. Then $y(a)$ is the presheaf with $y(a)(x)$ a one-point set when $x \le a$ and empty otherwise, so $y$ sends $a$ to the down-set $\set{x : x \le a}$. The map $a \mapsto \set{x : x \le a}$ is the classical embedding of a preorder into its down-sets ordered by inclusion.

Non-example: the assignment $A \mapsto \Hom(A, -)$ is not a functor $\mathcal{C} \to [\mathcal{C}, \mathbf{Set}]$; a morphism $f \colon A \to B$ induces $\Hom(B, -) \to \Hom(A, -)$, which goes the wrong way, so clause 2 fails for this assignment.