Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 3 / a / Solution

Write $H_c=\mathcal{C}(-,c)$. The category of elements $\int P$ of a categorical presheaf $P$ has objects $(c,x)$ with $x\in P(c)$; a morphism $(c,x)\to (d,y)$ is $u:c\to d$ satisfying $P(u)y=x$. It is a small category because $\mathcal{C}$ is small and each $P(c)$ is a set. Consider the diagram sending $(c,x)$ to $H_c$ and $u$ to the natural transformation given by postcomposition with $u$.

There is a canonical cocone ${\alpha }_{c,x}:H_c\to P$ withFunctoriality makes it a natural transformation, and the equation $P(u)y=x$ makes these transformations a cocone.

To prove its universal property, let $Q$ be any categorical presheaf with a compatible cocone ${\beta }_{c,x}:H_c\to Q$. DefineFor $u:c\to d$ and $y\in P(d)$, compatibility with the arrow $(c,P(u)y)\to (d,y)$ givesThus $t:P\to Q$ is natural. Naturality of each ${\beta }_{c,x}$ shows $t{\alpha }_{c,x}={\beta }_{c,x}$. Conversely any transformation with these equations must have the displayed values $t_c(x)$, so it is unique. We have proved the canonical colimit presentation of a presheaf: