There is something very weird with the way Hartshorne defines right derived functors.
Hartshorne p 204 Let $\mathfrak A$ be an abelian category with enough injectives, and let $F \colon \mathfrak A \to \mathfrak B$ be a covariant left exact functor. Then we construct the right derived functors $R^iF, i ≥ 0$, of $F$ as follows. For each object $A$ of $\mathfrak A$, choose once and for all an injective resolution $I^.$ of $A$. Then we define $R^iF(A) = h^i(F(I^.))$
With Prop 1.2A. Hartshorne claims that one can also compute the right exact functor $R^iF$ with a resolution $J^i$ that is acyclic for F.
But then, in prop. 2.6 (and similarly in prop. 8.4) Hartshorne states what was already true by definition: that the derived functors of $\Gamma(X,^.)$ coincide with the cohomology functors $H^i(X,^.)$. The proof goes around a circle to show the obvious fact that an injective resolution is equivalent to itself. Namely, that every injective is flasque (2.4), and flasques are acyclic (2.5), so by 1.2A. (above) we can use this very injective resolution(!).
What do we need to go around like that?