In Orbifolds as Groupoids there is the notion of an equivalence $\phi: H \rightarrow G$ between Lie groupoids (2.4) and of $G$-spaces (5.1).
Given a smooth functor $\phi: H \rightarrow G$ we can obtain a functor $$ \phi^*: G\text{-spaces} \rightarrow H\text{-spaces} $$ which sends a $G$-space $E$ with anchor $\pi:E \rightarrow G_0$ to the $H$-space $E \mathbin{_\pi \times_{\phi_0}} H_0$ with the induced action.
I'm trying to prove the following, stated in section 5.1:
If $\phi$ is an equivalence (2.4) then this functor $\phi^*$ is an equivalence of categories.
I'm struggling to come up with an inverse functor, that is given a $H$-space $F$ I need to find a $G$-space $E$ such that $\phi^*E$ is isomorphic to $F$.
Let $\pi: F \rightarrow H_0$ be the anchor map. My first attempt was to try to turn $F$ into an $E$-space with anchor map $\phi_0 \circ \pi$, but then there is no clear way to give $F$ a $G$-action.
My next thought was some sort of fibre product of $F$ with $G_0$ or $G_1$, allowing us to act by an element of $G_1$. Or even a product involving $H$ to make use of the following diffeomorphism in the definition of an equivalence: $$ H_1 \rightarrow (H_0 \times H_0) \times_{G_0 \times G_0} G_1, \quad h \mapsto (s(h), t(h), \phi_1(h)). $$ Basically I've tried lots of things and nothing has come out. Maybe the proof is more abstract and a direct construction isn't obvious.
I'd appreciate any hints or solutions anyone can offer!