Let $X$ and $Y$ be $G$-spaces for a group G (you may assume Hausdorff). Let $Y^X$ be the space of continuous mappings from $X$ to $Y$ with the compact open topology. This space carries a conjugation action of $G$.
It is often stated that under some minor regularity assumptions (X locally compact ?), this becomes a G-space, i.e. the action becomes continuous, but I have never seen a proof of this basic fact anywhere. I'd appreciate any hints!
If the spaces $G,X,Y$ are "nice", then you have a homoemorphism $$ \text{Hom}(G\times Y^X , Y^X )\cong \text{Hom}(G\times Y^X \times X, Y) .$$ So if you want to show that some map is contained in the set on the left side, you can also show that the map is contained in the right side.
Now what is the map you are looking for corresponding to? It should be a compistion of product maps of the $G$-action on $X$ the evaluation and the inverse $G$-action on $Y$.
So you need to ensure that the evaluation is continuous and the inverse map of $G$ is also continuous.
