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This is a problem from Hartshorne's Algebraic Geometry Chapter III Ex.6.4:

Let $X$ be a noetherian scheme, and suppose that every coherent sheaf on $X$ is a quotient of a locally free sheaf. Then for any $mathcal{G} in mathcal{Mod}(X)$, show that the $delta$-functor $mathscr{Ext}^{i}(cdot,mathcal{G})$ from $mathcal{Coh}(X)$ to $mathcal{Mod}(X)$, is a contravariant universal $delta$-functor.

My attempt and failure:

By the hint from this book, we want to show that this $delta$-functor is coeffacable. Now by condition we have a surjection from a locally free sheaf $mathcal{F'}$ to $mathcal{F}$. From Hartshorne's book we know that locally free sheaf of finite rank has vanished higher Ext sheaves, so I want to find a locally free sheaf of finite rank of $mathcal{F'}$. By extension of coherent sheaf we can find a coherent subsheaf of $mathcal{F'}$. But it seems that this sheaf not necessary becomes locally free again, and neither has a vanished higher Ext sheave from a priori.

Any help is appreciated.

Edit: I've thought about that Hartshorne implies the locally free sheaf to have finite rank since he also refers it as "$mathcal{Coh}(X)$ has enough locally frees". If this is the case I appreciate a counter example of the original statement above.