At the page 204 , Hartshorne defined right derived functors as $R^{i}F(A)$ = $h^i(F(I^{.}))$ , where $I^{.}$ is an injective resolution of $A$. Now on page 205 at Prop 1.2A he gives a natural isomorphism, $R^{i}F(A) \cong h^i (F(J^{.})) $, where $J^{.}$ is an $F$-acyclic resolution of $A$. Wouldn't it be just followed by the construction of right-derived functors given on page 204?
It is a natural isomorphism, that we are getting an additional fact here but is it the only thing here? or am I missing any crucial points?