Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 1 / i / Solution

Write $H_A=\mathcal{C}(-,A)$ for a representable presheaf. The contravariant Yoneda lemma states that, for every presheaf on a category $X$, evaluation at the identity is a bijectionnatural in both $A$ and $X$. Smallness of $\mathcal{C}$ ensures the relevant families and natural-transformation collections are sets.

For $x\in X(A)$ define ${\alpha }_B^x(h)=X(h)(x)$ for each $h:B\to A$. This family is a natural transformation: for $u:B^{\prime }\to B$, $X(u){\alpha }_B^x(h)=X(u)X(h)x=X(hu)x={\alpha }_{B^{\prime }}^x(hu)$. Its identity value is $x$. Conversely, naturality of any $\alpha$ at $h:B\to A$ gives ${\alpha }_B(h)=X(h){\alpha }_A(1_A)$. Thus evaluation and the displayed construction are inverse.

For a map $v:A\to A^{\prime }$, precomposing a transformation $H_{A^{\prime }}\to X$ with postcomposition $H_A\to H_{A^{\prime }}$ corresponds to applying $X(v):X(A^{\prime })\to X(A)$. For a transformation $\tau :X\to Y$, postcomposition corresponds to $x\mapsto {\tau }_A(x)$. These identities prove the stated naturality in both variables.