Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 5 / ii

Working with a small category $\mathcal{C}$ (or in a fixed ambient universe), for a categorical presheaf $X$ and fixed $U$, apply the categorical coend construction to $T(A,B)=\mathcal{C}(U,B)\times X(A)$. It is the quotient of the disjoint union of pairs $(h:U\to W,x\in X(W))$ by the relationsDefine ${\Phi }_U([h,x])=X(h)x$. It respects every relation because $X(fh)x=X(h)X(f)x$. Define ${\Psi }_U(y)=[1_U,y]$. Then ${\Phi }_U{\Psi }_U(y)=y$. Conversely the relation with $f=h:U\to W$ gives $[h,x]=[1_U,X(h)x]$, so ${\Psi }_U{\Phi }_U$ is the identity morphism. ThereforeFor $k:V\to U$, restriction on the categorical coend sends $[h,x]$ to $[hk,x]$, and ${\Phi }_V([hk,x])=X(k){\Phi }_U([h,x])$. A natural transformation $X\to Y$ likewise acts on the second coordinate and commutes with $\Phi$. Thus the density formula for presheaves is a natural isomorphism, not just a family of unrelated bijections.