Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 1 / i / Solution

For a categorical presheaf $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ and $A\in \mathcal{C}$, the Yoneda lemma gives a natural bijectionTo construct its inverse, take $x\in X(A)$ and set ${\theta }_U^x(h)=X(h)x$ for $h:U\to A$. For $k:V\to U$, the functor law gives $X(k){\theta }_U^x(h)=X(hk)x={\theta }_V^x(hk)$, so ${\theta }^x$ is a natural transformation. Conversely, naturality of any $\theta$ at $h:U\to A$ gives ${\theta }_U(h)=X(h){\theta }_A(1_A)$. Thus the two constructions are inverse, proving the bijection.

The bijection is natural in $X$: a natural transformation $t:X\to Y$ sends the distinguished element $x$ to $t_A(x)$ on either side. It is natural in $A$: for $f:A\to B$, precomposing a transformation $\mathcal{C}(-,B)\to X$ with postcomposition by $f$ sends its distinguished element $x\in X(B)$ to $X(f)x$. These equations prove the claimed naturality, rather than just an objectwise correspondence. Reversing morphisms gives the covariant form $\mathrm{Nat}(\mathcal{C}(A,-),Z)\cong Z(A)$ for $Z:\mathcal{C}\to \mathbf{Set}$.