I'm looking for a reference for the following result:

Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over $X$. Then the category of sheaf of $\mathcal{R}$-module over $X$ is equivalent to the category of non-degenerate $\Gamma_c(\mathcal{R})$-modules, where $\Gamma_c(\mathcal{R})$ is the ring of continuous compactly supported functions on $X$ with value in $\mathbb{R}$.

I would also be interested in references in the case where $X$ is compact or under more general assumption (like when $\mathcal{R}$ is replaced by any c-soft sheaf of rings...)