Hi,
is there something like :
  $X^{Y \times Z}$ homeomorph to $(X^Y)^Z $ with compact open topologies
if and only if "...."
with conditions "...." on Z ,Y and maybe X?
famous is : Z  T2 and Y locally compact =>  $X^{Y \times Z}$ homeomorph to $(X^Y)^Z $ and you can weak this a bit but not to an equivalence.