# Functors preserving (commuting with) exponentials

I have been unable to find any established names for functors preserving exponential objects in general ($F$ such that $F(A^B) \cong FA^{FB}$) and/or those "commuting" with functors $-^A$ (some functor $F$ such that for all objects $A$ one has $F \circ -^A \cong -^{FA} \circ F$).

Are there any such names, or am I just being stupid and missing something obvious (something along the lines of exponentials being preserved by those functors preserving limits).

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For cartesian closed categories, we have a notion of cartesian closed functor, which are functors which preserve the cartesian closed structure. In particular, they preserve exponential objects.

Functors preserving limits need not preserve exponentials, just as continuous lattice homomorphisms need not preserve the Heyting relative pseudocomplement.

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Thanks, I didn't think to search for such a thing. Also the fact that the n-lab page doesn't seem to mention any names for functors preserving just the exponentials suggests, to me, that no such established name exist (thus motivation the "accept"). –  Tilo Wiklund Sep 14 '11 at 17:02