Trivialising homeomorphisms for a principal $G$-bundle as $G$-space morphisms

Question

Let $P$ by a principal $G$-fibre bundle over a locally-compact Hausdorff space $X$. Denote by $$ h: U \times G \to P|_{U} $$ a trivialising homeomorphism for a trivialising open set $U \subseteq X$. Now the domain and range of $h$ are both $G$-spaces in an obvious way. Is $h$ a morphism of $G$-spaces? If so why? If true and not evident, I would be grateful for a reference.

Answer 1

I am not sure which definitions you use, but this is more or less true by definition. I suppose you would start by requiring the transition functions between the trivializations to be given by left multiplications in the group. Then you define the principal right action on $P$ by requiring that in the trivialization it is given by multiplication from the right in $G$. But this exactly means that the map $h$ you consider is $G$-equvariant for $(x,g)\cdot h:=(x,gh)$ and the principal right action on $P|_U$.