= 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 $B$, proving
$$
\boxed{F\cong\operatorname*{colim}_{(C,x)\in E^{\mathrm{op}}}\mathcal C(C,-).}
$$
Smallness 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.
Back to article page
