This is Theorem 1.10, pg5
(Kan) Let $A$ be a small category, together with a locally small category $C$ with has small colimits. For any functor $u:A \rightarrow C$, the evaluation at $u$,
$$u^*:C \rightarrow \hat{A}, \quad Y \mapsto u^*(Y):(a \mapsto Hom(u(a),Y).$$
has a left adjoint $u_{!}:\hat{A} \rightarrow C$. Moreover, there is a unique natural isomorphism
$$u(a) \simeq u_{!}(h_a), a \in ob(A). $$
It begins with
For each presheaf $X$ over $A$, we choose a colimit of the functor
$$A/X \rightarrow C, (a,s) \mapsto u(a). $$
which we denote by $u_{!}(X)$.
$A/X$ is the category of elements of $X$ defined on pg4. I do not understand the proof where it says
We have a canonical isomorphism $u(a) \simeq u_{!}(h_a)$ since $(a,1_a)$ is a final object of $(A,h_a)$.
Best Answer
It's just an application of the fact that if a category $\mathcal{J}$ has a terminal object $1$, then the colimit of any functor $F:\mathcal{J}\to \mathcal{C}$ is just $F(1)$, with the cocone defined by the maps $F(\tau_j):F(j)\to F(1)$. For a proof of this fact, see this question; alternatively, if you know about final functors, you can show that the inclusion functor $\mathbf{1}\to\mathcal{J}$ that takes the unique object of $\mathbf{1}$ to the terminal object of $\mathcal{J}$ is final.