When is the derived functor of a representable functor representable?

Question

Let $\mathcal{C}$ be an abelian category and $\mathcal{F}:\mathcal{C}\rightarrow \text{Set}$ a representable sheaf. Consider the $i$-th derived functor $R^i\mathcal{F}$ of $\mathcal{F}$. My question is when is this functor representable. Since by assumption there exists some element $X$ such that $\mathcal{F}\cong \text{Hom}(X,-)$, we have $R^i\mathcal{F}\cong \text{Ext}^i(X,-)$. For every exact sequence in $\mathcal{C}$ $$0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ we get an exact sequence $$\text{Ext}^{i-1}(X,C)\rightarrow \text{Ext}^{i}(X,A)\rightarrow \text{Ext}^{i}(X,B)\rightarrow \text{Ext}^{i}(X,C).$$ Since representable functors are left exact, that would mean that $\text{Ext}^{i-1}(X,-)=0.$ My questions is if this is sufficient for $R^i\mathcal{F}$ to be representable, and if not, what other restrictions on $F$ or equivalently $X$ can be impossed to make $R^i\mathcal{F}$ representable?