Homotopy-invariance of sheaf cohomology for locally constant sheaves

I succeeded in proving it myself:

Suppose we have homotopy inverses $f: X \to Y$ and $g: Y \to X$, where $X$ and $Y$ are locally connected and semi-locally simply connected (e.g. manifolds). Let $\mathcal{C}$ be a local system (a locally constant sheaf) over $Y$.

The adjoint equivalence of categories between sheaves and etale spaces restricts to an adjoint equivalence between local systems and covering spaces (this works for sheaves of sets, groups, modules etc.)

Then by homotopy invariance of fibre bundle pullbacks (on nice spaces), $f^*$ and $g^*$ will be homotopy inverses (i.e. giving an equivalence of categories) between the categories of covering spaces over $X$ and $Y$, and hence between the corresponding categories of local systems of these two spaces.

Without loss of generality, lets take $g^*$ to be the right adjoint to $f^*$ (since an equivalence of categories can be promoted to an adjoint equivalence).

Then using the natural isomorphisms $\Gamma(Y, \mathcal{C}) \cong \mathrm{Hom}(\mathbb{1}, \mathcal{C})$ and $f^{-1} \mathbf{1} \cong \mathbf{1}$ where $\mathbf{1}$ is the tensor unit (e.g. the constant sheaf of integers if we work with sheaves of abelian groups), we get natural isomorphisms $$\Gamma(Y, \mathcal{C}) \cong \mathrm{Hom}(\mathbf{1}, \mathcal{C}) \cong \mathrm{Hom}(\mathbf{1}, g^{-1}f^{-1}\mathcal{C}) \cong \mathrm{Hom}(f^{-1}\mathbf{1}, f^{-1}\mathcal{C}) \cong \mathrm{Hom}(\mathbf{1}, f^{-1}\mathcal{C}) \cong \Gamma(X, f^{-1}\mathcal{C}).$$

The (homology of the) right derived functor $R\Gamma(X, -)$ is sheaf cohomology, so we are done.