Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 26 / 3 / Solution

The comma category $(A\downarrow F)$ has objects $(C,u)$ with $C\in \mathcal{C}$ and $u:A\to FC$ a set map. A morphism $(C,u)\to (D,v)$ is an arrow $h:C\to D$ such that $Fh\circ u=v$. Composition and identities are those of $\mathcal{C}$. For $A=1$, this is the category of elements $E$ of $F$: objects are $(C,x)$ with $x\in FC$ and arrows satisfy $Fh(x)=y$.

Associate to $(C,x)$ the covariant representable functor $H_C=\mathcal{C}(C,-)$. An arrow $h:(C,x)\to (D,y)$ of $E$ gives, by precomposition, $H_D\to H_C$. Thus this is a diagram $H:E^{\mathrm{op}}\to [\mathcal{C},\mathbf{Set}]$. There is a compatible natural map $H_C\to F$ sending $f:C\to B$ to $Ff(x)$.

Colimits in this functor category are pointwise. At an object $B$, the induced map from the colimit sends a class represented by $(C,x,f:C\to B)$ to $Ff(x)$. It is surjective because $z\in FB$ is represented by $(B,z,1_B)$. It is injective because every representative with value $z$ is identified with that same canonical representative: the arrow $f:(C,x)\to (B,z)$ in $E$ becomes an arrow of the indexing diagram sending $1_B$ to $f$. These bijections are natural in $B$, provingSmallness of $\mathcal{C}$ and set-valuedness of $F$ ensure that $E$ is small, so the displayed colimit is defined. This proves the introductory density request before the three equivalent conditions.