Yoneda lemma

Remarks

Proof idea: let $\alpha$ be natural and $x = \alpha_A(\id_A)$. For $f \colon X \to A$, naturality applied to $f$ and the element $\id_A$ gives $\alpha_X(\id_A \circ f) = F(f)(\alpha_A(\id_A))$, that is, $\alpha_X(f) = F(f)(x)$. So $\alpha$ is determined by $x$, and a direct check shows that every $x$ defines a natural transformation in this way.

Example: take $F = \Hom(-, B)$ for an object $B$. The lemma says that natural transformations $\Hom(-, A) \to \Hom(-, B)$ correspond to elements of $\Hom(A, B)$, that is, to morphisms $A \to B$; each one is post-composition with that morphism.

The covariant form is the same: for a functor $G \colon \mathcal{C} \to \mathbf{Set}$, natural transformations $\Hom(A, -) \to G$ correspond to elements of $G(A)$. Local smallness is needed for $\Hom(-, A)$ to take values in $\mathbf{Set}$.

Going deeper

For a student

The lemma is the reason that the Yoneda embedding $A \mapsto \Hom(-, A)$ is fully faithful, so every category sits inside a category of presheaves without loss of information. It implies that a representing object of a representable functor is unique up to unique isomorphism. This is why objects defined by a universal property are unique up to unique isomorphism. In practice it is used to prove that two objects are isomorphic by showing that their hom functors are naturally isomorphic. Applied to a group seen as a one-object category, it gives Cayley's theorem, that every group embeds in a group of permutations of its own elements.