Assume a topos $\mathcal{S}$ as the base topos, and we work in this topos as in naive set theory (without choice or excluded middle). Take a Grothendieck topos $\mathcal{E} \to \mathcal{S}$ with a site of definition $\mathcal{C}$. As usual in the literature consider that $\mathcal{C}$ has objects, and that these objects are objects of $\mathcal{E}$ which are generators in the sense that given any $X \in \mathcal{E}$ the family of all $f: C \to X$ all $C$ in $\mathcal{C}$ is epimorphic.

Consider $F: \mathcal{E} \to \mathcal{S}$ to be the inverse image of a point. Then the family $Ff: FC \to FX$ is epimorphic in $\mathcal{S}$. My question is the following:

Can I do the following ? (meaning, is it correct the following arguing, certainly valid if $\mathcal{S}$ is the topos of sets):

Given $a \in FX$, take $f: C \to X$ and $c \in FC$ such that $a = Ff(c)$

This is related to the validity in $\mathcal{S}$ of the following:

Given $x \in \coprod_{i \in I}{S_i}$, then $x \in S_i$ for some $i \in I$.