There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?

That is, for each small abelian category $A$ does there exist a small stable $\infty$-category $\underline A$ and a homological functor $\underline A \to A$ such that every homological functor $C \to A$ (where $C$ is a small stable $\infty$-category) factors uniquely through $\underline A \to A$ via an exact $\infty$-functor $C \to \underline A$?

On the other hand, for each small stable $\infty$ category $C$, does there exist an abelian category $\overline C$ and a homological functor $C \to \overline C$ such that every homological functor $C \to A$ (where $A$ is a small abelian category) factors uniquely through $C \to \underline C$ via an exact functor $\underline C \to A$?