Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left. Consider the simplicial space $[G\backslash X]_{\cdot}$ where $$ [G\backslash X]_n=\underbrace{G\times \ldots \times G}_{n\text{ copies of }G\text{'s}}\times X $$ with structural maps $$ d_0(g_1,\ldots, g_n,x)=(g_2,\ldots,g_n,g_1^{-1}x); $$ $$ d_i(g_1,\ldots, g_n,x)=(g_1,\ldots, g_ig_{i+1},\ldots, g_n,x), ~1\leq i\leq n-1; $$ $$ d_n(g_1,\ldots, g_n,x)=(g_1,\ldots, g_{n-1},x); $$ and $$ s_0(g_1,\ldots, g_n,x)=(e,g_1,\ldots, g_n,x); $$ $$ s_i(g_1,\ldots, g_n,x)=(g_1,\ldots, g_i,e , g_{i+1},\ldots, g_nx),~1\leq i\leq n-1; $$ $$ s_n(g_1,\ldots, g_n,x)=(g_1,\ldots, g_n,e,x). $$
A $G$-equivariant sheaf on $X$ is a pair $(\mathcal{F},\theta)$, where $\mathcal{F}\in \text{Sh}(X)$ and $\theta$ is an isomorphism $$ \theta: d_0^*\mathcal{F}\overset{\sim}{\to} d_1^*\mathcal{F}, $$ satisfying the cocycle condition $$ d_2^*\theta\circ d_0^*\theta=d_1^*\theta, \text{ and } s_0^*\theta=\text{id}_{\mathcal{F}}. $$ We denote the category of $G$-equivariant sheaves on $X$ by $\text{Sh}_G(X)$.
If the $G$-action on $X$ is free, then we should have a category equivalence $$ \text{Sh}(G\backslash X)\overset{\sim}{\to}\text{Sh}_G(X). $$
This result seems to be well-known and we do not require $G$ is compact.
My question is: how to prove this equivalence? Is there a reference with detailed proof?