# 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).