What are the minimal conditions on three topological spaces $X,Y$ and $Z$ such that with the compact-open topology the map

$${(X^Y)}^Z \to X^{Y \times Z}$$

given by taking adjoints is a homeomorpism. The map sends $f: Z \to X^Y$ to $g:Y \times Z \to X$ by the relation $g(y,z)=f(z)(y)$.

This result is known for $Z$ Hausdorff and $Y$ locally compact. I'm interested in a proposition of the form the adjoint construction is a homeomorphism of mapping spaces if and only if some statement regarding the spaces $X$, $Y$ and $Z$.