We equip sets of continuous functions $C(X,Y)$ with compact-open topologies, which are topologies generated by prebasis of sets of form $$ W(K,U) = \{ f \in C(X,Y) : f(K) \subset U\}, $$ where $K$ is a compact subset of $X$ and $U$ is an open subset of $Y$.
The problem is to show that spaces with such topology $C(X,Y)\times C(X,Z)$ and $(X,Y \times Z)$ are homeomorphic.
I want to show that map
$$\varphi : C(X,Y)\times C(X,Z) \to C(X,Y \times Z)$$
$$ \varphi : (f,g) \mapsto f \times g $$
$$ \varphi(f,g)(x) := (f(x),g(x)) $$
is a homeomorphism.
From the universl property of product we can infer that $\varphi^{-1}$ exists and continuous.
On the other hand I don't know how to show, that $\varphi$ is continuous. Here is my work so far:
$C(X,Y \times Z)$ has a topology generated by sets of the form $W(K,U)$, where $K$ is compact and $U = \bigcup_{i \in I} V_i \times W_i$, with $V_i$ and $W_i$ being open in $Y$ and $Z$ respectively. If $f \in W(K,U)$, then $f(K)$ is compact as $f$ is continuous, which means that $f(K)$ can be covered by a finite set of rectangles $\; (V_{i_j} \times W_{i_j})^n_{j=1}$. So I want to claim that $f$ has an open neighborhood in $W(K,U)$ of of the form $ \bigcap_{(k,j) \in J }W(C_k, V_{i_j} \times W_{i_j} )$ for some finite set $J$ and a collection of compacts $C$. But I don't know how to construct them.
I noticed that it is possible to construct compact sets
$$ \alpha_k = f^{-1}\left(Y \times Z \setminus \bigcup^n_{j = 1 , j \neq k} V_{i_j} \times W_{i_j}\right) \cap K, $$ But they are insufficient as they cover only points in $x \in K$, such that $f(x)$ does not belong to any intersection of said open rectangles. I don't know how to construct compacts which will correspond to intersections.
I know how to prove this result in case $X$ is Hausdorff, but I heard that it is also true in general.
To see the continuity of $\varphi$: let $W(K, U \times V)$ be a subbasis element of $C(X, Y \times Z)$, (where $U \subseteq Y$ and $V \subseteq Z$ are open) and suppose $(f,g) \in O:=\varphi^{-1}[W(K,U \times V)]$. We want to show $(f,g)$ is an interior point of $O$.
(Lemma 3.4.6 in Engelking (General Topology, 2nd ed.) says that in a compact-open $C(X,Y)$ we can take all sets $W(K,O)$ as a subbase where we choose the $O$ not all opens but from a fixed subbase for $Y$, this is handy to reduce the number of open sets to consider in proofs.)
Note that $f \in W(K, U)$ and $g \in W(K,V)$ and that the open neighbourhood $W(K,U) \times W(K,V)$ is mapped by $\varphi$ into $W(K, U \times V)$, as is easily seen, so indeed $(f,g) \in W(K,U) \times W(K,V) \subseteq O$, as required.