TITLE: Functorial properties of the compact open topology.
QUESTION [6 upvotes]: Let $X,Y,Y'$ be topological spaces and $A\subseteq Y$ a subspace. Every set of continuous maps is equipped with the compact-open topology.

Is the canonical map $\mathcal{C}(X,Y)\times\mathcal{C}(X,Y')\rightarrow\mathcal{C}(X,Y\times Y')$ a homeomorphism?
Does $\mathcal{C}(X,A)$ carry the subspace topology of $\mathcal{C}(X,Y)$, i.e. is the map $\mathcal{C}(X,A)\rightarrow \mathcal{C}(X,Y)$ given by postcomposing with the inclusion a homeomorphism on its image?

If it changes anything, all spaces can be assumed to be locally compact, Hausdorff and paracompact.
I think the first claim should follow from the fact that $\mathcal{C}(X,\_)$ is a right adjoint of $X\times\_$, so preserves limits, in particular products. (At least if $X$ is locally compact.)  Is the reasoning correct? What about the second question?

REPLY [3 votes]: Yes, as noted in comments, your argument works.
If $K$ is a compact subset of $X$ and $U$ is open subset of $Y$, then the inclusion $A\subset Y$ maps $\{f\in C(X,A):f(K)\subset A\cap U\}$ to the $\{f\in C(X,A):f(K)\subset U\}\bigcap C(X,A)$. So the map $C(X,A)\to C(X,Y)$ identifies elements of subbase of own compact-open topology of $C(X,A)$ with elements of subbase of induced topology on it.