Affine pushforward is compatible with sheaf cohomology

Apply the Godement construction of [F1] to the underlying abelian sheaf of $\mathcal{F}$, retaining its module structure at every stage. Explicitly, for an ${\mathcal{O}}_X$-module $\mathcal{G}$, the first Godement term has $C^0(\mathcal{G})(V)={\prod }_{x\in V}{\mathcal{G}}_x$, with $a\in {\mathcal{O}}_X(V)$ acting in the $x$-coordinate through $a_x\in {\mathcal{O}}_{X,x}$. The germ map $\mathcal{G}\to C^0(\mathcal{G})$ is ${\mathcal{O}}_X$-linear; take its cokernel in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ and repeat. By [F9] the resulting kernels and cokernels have the same underlying abelian sheaves as in the abelian-sheaf construction, so this gives an exact resolution $0\to \mathcal{F}\to G^{\bullet }$ by ${\mathcal{O}}_X$-modules whose underlying abelian complex is the Godement resolution. Every $G^p$ is flasque and $H^q(X,\mathcal{F})\cong H^q(\mathrm{\Gamma }(X,G^{\bullet }))$ naturally in $\mathcal{F}$ by [F1].

For every $p$ the sheaf $f_{\ast }{G}^p$ is flasque: by [F3] its sections on an open $V\subseteq S$ are those of $G^p$ on $f^{-1}V$, and for $V^{\mathrm{\prime }}\subseteq V$ the restriction map of $f_{\ast }{G}^p$ is the restriction map of $G^p$ along the open inclusion $f^{-1}{V}^{\mathrm{\prime }}\subseteq f^{-1}V$, which is surjective by [F2].

Every flasque sheaf $\mathcal{G}$ on $X$ is $f_{\ast }$-acyclic, that is $R^q{f}_{\ast }\mathcal{G}=0$ for all $q>0$. Indeed, let $V\subseteq S$ be affine; then $f^{-1}V$ is affine by [F6] and $H^q(f^{-1}V,\mathcal{G})=0$ for $q>0$ by [F4]. The local-section formula [F5] exhibits $R^q{f}_{\ast }\mathcal{G}$ as the sheafification of the presheaf $V\mapsto H^q(f^{-1}V,\mathcal{G})$, and as in the proof of [F8] its stalks are filtered colimits over the affine opens of $S$, all of whose values vanish; hence these stalks are zero and $R^q{f}_{\ast }\mathcal{G}=0$.

Apply the acyclic-resolution theorem [F7] to the left exact functor $f_{\ast }$ and the resolution $G^{\bullet }$ of $\mathcal{F}$, all of whose terms are $f_{\ast }$-acyclic by step 1.3: there are canonical isomorphisms $R^q{f}_{\ast }\mathcal{F}\cong H^q(f_{\ast }{G}^{\bullet })$ for every $q\ge 0$. Since $f$ is affine and $\mathcal{F}$ quasi-coherent, [F8] gives $R^q{f}_{\ast }\mathcal{F}=0$ for $q>0$; therefore the complex $f_{\ast }{G}^{\bullet }$ is a resolution of $f_{\ast }\mathcal{F}$ by flasque sheaves, the degree-zero cohomology being $H^0(f_{\ast }{G}^{\bullet })=f_{\ast }\mathcal{F}$ by left exactness of $f_{\ast }$.

Apply [F7] to the global-sections functor ${\mathrm{\Gamma }}_S$ and the resolution $f_{\ast }{G}^{\bullet }$ of $f_{\ast }\mathcal{F}$ from step 2.1, whose terms are flasque, hence ${\mathrm{\Gamma }}_S$-acyclic by [F4]: there are canonical isomorphisms $H^q(S,f_{\ast }\mathcal{F})\cong H^q(\mathrm{\Gamma }(S,f_{\ast }{G}^{\bullet }))$ for every $q\ge 0$. Since $\mathrm{\Gamma }(S,f_{\ast }{G}^p)=G^p(X)=\mathrm{\Gamma }(X,G^p)$ by [F3], the right-hand cohomology is $H^q(\mathrm{\Gamma }(X,G^{\bullet }))$, which is $H^q(X,\mathcal{F})$ through the natural isomorphism of step 1.1. Composing gives an isomorphism $H^q(S,f_{\ast }\mathcal{F})\to H^q(X,\mathcal{F})$ for every $q\ge 0$.

Naturality and canonical identification. The Godement construction in step 1.1 is functorial in $\mathcal{F}$, and the acyclic-resolution comparison isomorphisms in steps 2.1 and 3.1 are natural by [F7]. Their composite is therefore the canonical map obtained by evaluating the same complex first as ${\mathrm{\Gamma }}_S(f_{\ast }{G}^{\bullet })$ and then as ${\mathrm{\Gamma }}_X(G^{\bullet })$ under the identity of functors ${\mathrm{\Gamma }}_S\circ f_{\ast }={\mathrm{\Gamma }}_X$; in degree zero it is the identity on $\mathcal{F}(X)$. Boundary cases: if $X=\mathrm{\varnothing }$ or $\mathcal{F}=0$ then $f_{\ast }\mathcal{F}=0$ and both sides vanish; if $S=\mathrm{\varnothing }$ both sides are zero sheaves on the empty space; if $f$ is the identity then the comparison is the identity map. The Axiom of Choice is inherited from [F1] and [F4], and Dependent Choice from [F7]; no other selection is made.