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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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    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!2012-02-16

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