Representing homotopy as a continuous path in non-locally compact spaces

Question

Let us consider three topological spaces $X, Y, Z$. It is well known that a sufficient condition to establish the equivalence (homeomorphic spaces) between $C^{0}(Z\times X,Y)$ and $C^{0}\left(Z,C^{0}(X,Y)_{co}\right)$ (where "co" means equipped with the compact-open topology) is that $X$ is Hausdorff and locally compact. Therefore, these assumptions do not cover the case where $X$ is an infinite-dimensional Hilbert space because it lacks local compactness. Does this result remain true in the case of $$ C^{0}(Z\times H,H)\overset{?}{\simeq}C^{0}\left(Z,C^{0}(H,H)_{co}\right) $$ where $H$ is an infinite dimensional Hilbert space? Should we consider other topologies for $C^{0}(H,H)$? My question is motivated by the fact that I would like to represent the homotopy $h\in C^{0}(\left[0,1\right]\times H,H)$ in this more general setting as a continuous path in $C^{0}\left(\left[0,1\right],C^{0}(H,H)_{co}\right)$.