Are there universal homological functors?

Question

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 $C \to \overline C$ via an exact functor $\overline C \to A$?

Answer 1

Here is half an answer (i.e. an answer to the last question). Given a small stable $\infty$-category $C$ and a small abelian category $A$, any functor $C\to A$ factors through the homotopy category $ho(C)$ of $C$. Hence the category of homological functors from $C$ to $A$ is equivalent to the category of homological functors from the triangulated category $ho(C)$ to $A$. There is a universal homological functor from $ho(C)\to A(C)$: this is documented in Verdier's thesis (see Théorème 3.2.1 in Astérisque 239, an electronic copy being available here), although Verdier thanks Freyd for sharing the construction with him. A modern reference in English is Theorem 5.1.18 in Neeman's book.