From Moerdijk, Classifying spaces and classifying topoi, page 22. Consider a right $G$-set $S$ with the discrete topology. Let $E$ be a principal $G$-bundle over the topological space $X$. One can construct another principal $G$-bundle $S\otimes_G E$ as the quotient of $S\times E$ by the equivalence relation $(s\cdot g,e)\sim(s, g\cdot e)$. Moreover, one can construct another right $G$-set substituting $E$ by the set $\tilde G$, which is $G$ with the right multiplication action (but which, being $G$, supports also left multiplication and so the relation is well defined). So consider an arbitrary topos morphism $f:Sh(X)\longrightarrow \mathcal BG$, where the domain is the topos of sheaves on $X$ and the codomain is the topos of right $G$-sets.
The aim is to prove that, for any $S$,
$$(S\otimes_G\tilde G)\otimes_G f^*(\tilde G)\cong f^*(S \otimes_G\tilde G);$$
from this it will follow that, since $S\otimes_G\tilde G\cong S$, then
$$S\otimes_G f^*(G)\cong f^*(S \otimes_G\tilde G)\cong f^*(S)$$
and so the functors $-\otimes_G f^*(G)$ and $f^*$ coincide.
Can someone give me a hint in order to prove the key isomorphism? I tried with universal properties, but I find some problems in dealing with maps from $G$-sets to sheaves, which are not very natural unless they are actually $f^*$...
Thank you in advance.