Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 4 / ii / Solution

For fixed $U$, the categorical coend on the right is the quotient of ${\coprod }_W\mathcal{C}(U,W)\times X(W)$ by the identificationsDefine ${\Phi }_U([h,x])=X(h)x$. This is well defined because the contravariant functor law gives $X(fh)x=X(h)X(f)x$. Its proposed inverse sends $y\in X(U)$ to $[1_U,y]$. The composite to $X(U)$ is the identity. In the other direction the coend relation, applied with $f=h$ and initial map $1_U$, gives $[h,x]=[1_U,X(h)x]$. Thus the inverse is genuine andFor $a:U^{\prime }\to U$, the coend map replaces $h$ by $ha$; its image is $X(ha)x=X(a)X(h)x$, proving naturality in $U$. A morphism of presheaves acts on $x$ and commutes with every $X(h)$ by naturality, proving naturality in $X$. This is the density formula for presheaves.