cocomplete implies geometric morphism

Question

How cocompleteness of a topos $\cal E$ implies that $\cal E$ is equipped with a geometric morphism $\gamma:{\cal E}\to {\text {Set}}$ ?

Answer 1

As mentioned in the comments, $G$-torsors are related to the constant sheaf: for a space $X$, there is a morphism $f^* = \Delta: \operatorname{Set} \to \operatorname{Sh}(X) $ which sends $S$ to the constant sheaf $\Delta S.$ Its right adjoint $f_*$ is given by global sections $P \mapsto P(X).$

The book you read (Mac Lane & Moerdijk) refers to the generalization of this construction to any cocomplete topos $E$, described on p.350: the "constant sheaf" functor $f^*: \operatorname{Set} \to E$ is given by $S \mapsto \coprod\limits_{s \in S}1$ wheareas its right adjoint is now given by $P \mapsto \operatorname{Hom}_E(1, P)$ (note that this gives global sections in the case when $P$ is a sheaf). Together, these constitute a geometric morphism $E \to \operatorname{Set}.$