In his Algebraic Geometry Robin Hartshorne shows in Proposition III.6.5 that if the category of $\mathcal{O}_X$-modules of a scheme $X$ has enough locally frees, then a locally free resolution may be used to compute the $\mathcal{Ext}^* (-,\mathcal{G})$ functors.
However, in Caution III.6.5.2 he then writes:
The results (6.4) and (6.5) do not imply that $\mathcal{Ext}$ can be construed as a derived functor in its first variable. In fact, we cannot even define the right derived functors of $\text{Hom}$ or $\mathcal{Hom}$ in the first variable because the category $\mathfrak{Mod}(X)$ does not have enough projectives.
But I thought that what Hartshorne proves is exactly that the class of locally free sheaves is adapted to the $\mathcal{Hom}(-,\mathcal{G})$ functor for any $\mathcal{O}_X$-module $\mathcal{G}$, and hence its right derived functor exists and may be computed using a locally free resolution.
Am I missing something, or is this just a consequence of Hartshorne only defining derived functors in the situation of enough projectives/injectives?