I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of $X$, have the compact open topology. I want to show that an action of $G$ on $X$, call it $\gamma_1:G\times X\rightarrow X$ is continuous iff its associated homomorphism $\gamma_2:G\rightarrow H(X)$, where $g\mapsto \phi_g$ (left-translation by $g$), is continuous.
I can do the $\Leftarrow$ direction: assuming $\gamma_2$ is continuous, the map $(g,x)\mapsto (\phi_g,x)$ is continuous. Since the evaluation map $(\phi_g,x)\mapsto \phi_g(x)$ is continuous, and $\gamma_1$ is the composition of these two, $\gamma_2$ is continuous.
The other direction is where I'm not sure how to proceed. We can take $\phi_g\in H(X)$ and a subbasis set $S(C,U)=\{f:f(C)\subseteq U\}$ for $C$ compact, $U$ open, such that $\phi_g\in S(C,U)$. So $\gamma_2^{-1}(S(C,U))=\{h\in G:h\cdot C\subseteq U\}$. This is all true, but I am not sure it is helpful. I am not sure what the right approach is; in particular I don't see how/when to leverage the fact that $\gamma_1$ is continuous.
Thanks for any hints, direction, insight, etc.