The FHHF theorem, on functors and cohomology

For (ii): from \(H^i(C^\bullet) = \ker\delta^i/\im\delta^{i-1}\) one has \(0 \to \im\delta^{i-1} \to \ker\delta^i \to H^i \to 0\), and applying the left exact \(F\) gives \(0 \to F(\im\delta^{i-1}) \to F(\ker\delta^i) \to FH^i\). Comparing with the corresponding sequence for \(F(C^\bullet)\) needs one intermediate fact: a left exact \(F\) commutes with kernels, giving a natural monomorphism \(\im F(f) \to F(\im f)\) via \(\coker F(f) \to F(\coker f)\), and \[ \im F(f) = \ker\qty(\coker F(f)) \to \ker\qty(F(\coker f)) \cong F(\ker \coker f) = F(\im f) . \] The resulting diagram induces the map by universal properties. Part (i) is the same argument on \(0 \to H^i \to \coker\delta^{i-1}\to \im\delta^i \to 0\) with the arrows reversed. For (iii), an exact \(F\) commutes with both kernels and cokernels, so \(F(\ker\delta^i) = \ker F(\delta^i)\) and \(F(\im\delta^{i-1}) = \im F(\delta^{i-1})\), and the quotients agree.

Exact functors commute with cohomology; for one-sided exact functors, the direction of the natural map is determined by which kernels or cokernels the functor preserves. The standard application is that filtered colimits are exact in \(\mods{A}\), so \(H^i\) commutes with filtered direct systems of sheaves — which is how one reduces cohomology computations to finitely generated pieces.