This is in page 11, Prop 6.1 of Mitchell's notes on Fibre bundles. I am quoting the proposition below.
Let $\pi:P \rightarrow B$ be a principal $G$ bundle, $X$ a right $G$-space. $Hom_G(P,X)$ denotes the set of $G$-equivariant maps, $\Gamma(P \times_G X \rightarrow B)$ denotes the sections of $q$, where $q: P \times_G X \rightarrow B$ is a fibre bundle map.
Given $ f \in Hom_G(P,X)$, this induces a map $P \rightarrow P \times X$ given by $p \mapsto (p,f(p))$. From UP where $B$ is regarded as $P/G$ of map $\pi$, there is an induced map $$s=s_f :B \rightarrow P \times_G X$$
There is a natural bijection $\phi:Hom_G(P,X) \rightarrow \Gamma(P \times_G X \rightarrow B)$ given by $f \mapsto s_f$.
The proof is to prove the case when $P=B \times G$ and use this categorial argument.
I believe this is an argument in category of Sets. But I am completely lost how the first map is the equalizer of the "second two" - how are there two maps?