Proposition 2

Let $(E, p)$ be a principal fibre space with base B and group G, and let $\varepsilon$ be the section $b\mapsto (b, e)$ of the trivial principal fibre space $(B\times G$, pr$_1)$, where $e$ denotes the identity element of G. The mapping $h\mapsto h\circ \varepsilon$ is a bijection of $\mathscr{C}_B^G(B\times G; E)$ onto the set $\mathscr{S}(B; E)$ of continuous sections of $p$. The inverse bijection associates to a continuous section $s$ of $p$ the B-morphism $(b, g)\mapsto s(b)\cdot g$.