If the category of elements of a presheave has terminal object, then it is representable.

Question

Let $F$ be a presheave on a category $C$. If $F$ is representable, then there is a $B \in C$ such that $F=Hom(-,B)$. Then, it is not hard to prove that its category of elements $E(F)$ is equivalent to the slice category $C_{/ B}$.

Is it true that if the category of elements $E(F)$ of $F$ has a terminal object, then $F$ is representable? If so, how do I prove it?

Answer 1

Yes, this is true.

Suppose that $E(F)$ has a terminal object $(B, y)$, and define $\theta : F \to \mathsf{y}(B)$ as follows. Given $A \in \mathrm{ob}(\mathcal{C})$, define $$\theta_A : FA \to \mathrm{Hom}_{\mathcal{C}}(A,B)$$ by letting $\theta_A(x)$ be the unique $f : A \to B$ given by the (unique) morphism $(A,x) \to (B,y)$ in $E(F)$.

Verify that $\theta$ is a natural isomorphism.

Answer 2

For a presheaf $F$ on $\mathcal C$, recall that $E(F)$ is defined as the category whose objects are $(c,x)$ where $c\in \mathcal C$ and $x\in F(c)$, and whose morphisms $(c,x) \to (c',x')$ are those $f:c\to c'$ in $\mathcal C$ such that $F(f)(x') = x$.

One consequence of the Yoneda lemma is that $$F \simeq \mathrm{colim}_{(x,c)\in E(F)} \mathsf y(c) \tag{1}$$ (where I use the same notation as Clive for the Yoneda embedding). Now suppose that there is a terminal object $(c_t,x_t)$ in $E(F)$: a colimit taken over a category with a terminal object is just the value of the diagram at that terminal object. Hence $F\simeq \mathsf y (c_t)$.

(Of course the detail of the proof of (1) should fall back to Clive's answer.)