A question on Hartshorne Chapter III Proposition 2.6

Question

When I read Hartshorne, I saw the Proposition 2.6 in Chapter III as follows:

Let $(X,\mathcal{O}_X)$ be a ringed space. Then the derived functors of the functor $\Gamma(X,-)$ from the category of $\mathcal{O}_X$-modules to the category of abelian groups coincide with the cohomology functors $H^{i}(X,-)$.

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I am pretty confused here because I think it is just the definition of cohomology functor... Why should we bother to use the result about the flasque sheaf and acyclic resolution to prove this?

So I must miss something important and obvious. Please point it out. Thanks!

Answer 1

The functor $\Gamma(X,-)$ is initially defined on page 207 as a functor $Sh(Ab, X) \to Ab$, where $Sh(Ab, X)$ is the category of sheaves of abelian groups on $X$.

You have the forgetful functor $U : Mod(O_X) \to Sh(Ab, X)$, and the claim is that $$R^i(\Gamma(X,-) \circ U) \cong R^i(\Gamma(X,-)) \circ U,$$ which does need a proof.