In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a cohomology theory on a manifold $M$, Warner (although the construction is by Cartan-Eilenberg) says that a fine torsionless resolution of a constant sheaf $\mathcal{H}=M\times K$,

$0\rightarrow\mathcal{H}\rightarrow S_{0}\rightarrow S_{1}\rightarrow S_{2}\rightarrow S_{3}\rightarrow\cdots$,

(where $S^{*}$ is a cochain complex of sheaves) defines canonically a cohomology theory on $M$, he then defines a sheaf cohomology group of a sheaf $\mathcal{F}$ in this cohomology theory as

$H^{q}(M,\mathcal{F})=H^{q}(\Gamma(\mathcal{S}^{*}\otimes\mathcal{F}))$,

here $\Gamma$ is the K-module of sections on the manifold. I don't know if I'm missing something here but this kind of baffled me as I thought the sheaf cohomology group of a sheaf was the right derived functor of the global sections functor, I don't understand what tensor products of sheaves are doing here quite frankly. I'm trying to use Cartan-Eilenberg's axiomatic sheaf theory construction to define an injective resolution of a constant sheaf on a SITE (with a topology given by a Grothendieck topology) and I'm practically trying to apply this axiomatic sheaf theory construction verbatim, so I don't know if I'm on the right path here or if there's a book I should refer to or something I should know, thanks.