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One can easily show by adjunction-nonsense that if we are given toposes $\cal E,F$ and a geometric morphism $f\colon \cal E\to F$ then there exist canonical arrows $$ \begin{gather} (f_*A)\times B\to f_*(A\times f^*B)\\ f^*(A\times f_*B)\to (f^*A)\times B \end{gather} $$ Working in categories of sheaves (for example $\mathcal O_X$-modules, $X$ a variety) one can also give sufficient condition (=flatness of one of the sheaves involved) for these maps to be isomorphisms. Can these condition be generalized to the scenario of Grothendieck and/or elementary toposes?

I'm interested in this because it seems to me that when they are iso, then $f$ commutes with the exponential bifunctor: $$f_*(B^A)\cong f_*(B)^{f_*(A)}$$ $$f^*(B^A)\cong f^*(B)^{f^*(A)}$$

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It sounds like you are interested in logical functors. – Zhen Lin Feb 16 '12 at 19:36
So are you saying that when $f$ is atomic, then $(f_*A)\times B\cong f_*(A\times f^*B)$ and $f^*(A\times f_*B)\cong (f^*A)\times B$? – tetrapharmakon Feb 16 '12 at 20:24
I'm not entirely sure, but I do remember that logical functors are supposed to preserve the logical structure of an elementary topos, and I think the exponentials are part of the logical structure. I may be wrong. – Zhen Lin Feb 16 '12 at 21:42
What I think I'm able to prove according to the page of nlab is the following: "$f_*(B^A)\cong f_*(B)^{f_*(A)}$ iff $f_*$ is an equivalence of categories" Which I find quite astonishing! – tetrapharmakon Feb 16 '12 at 23:53

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