In Caramello, Theories, Sites, Toposes, I read that
Every Grothendieck topos $\mathcal E$ admits a unique (up to isomorphism) geoemtric morphism $\gamma_{\mathcal E}:\mathcal E\to \bf Set$. The direct image of $\gamma_{\mathcal E}$ is the global section functor $Hom_{\mathcal E}:\mathcal E\to \bf Set$, while the inverse image functor $\gamma^*_{\mathcal E}$ is given by $S\to\coprod_{s\in S}1_\mathcal E$.
Now I have two questions:
1) if the topos is not cocomplete, does this definition make sense? I do not think that every sheaf topos over a small site is cocomplete.
2) Why is this unique? My guess is the following: by a well-known fact about topoi, it suffices to prove that the inverse image part is unique. But since it commutes with finite limits, then it should take the terminal object to the terminal object; hence, since it commutes with colimits and $\bf Set$ is generated under colimits by its terminal object, it follows that it must be as above. Am I right?
Thank you in advance.
