Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P $ an additive $k$-linear exact and faithful functor. I want to know when $F$ is equivalent to the forgetful functor from the category of co-modules over a co-algebra in $P$ to $P$. What are the known results in this direction?
A necessary condition for the existence of this equivalence is the existence of a "box product": $$\boxtimes : P\times M \to M,$$ which admits natural isomorphisms: $$F(A\boxtimes X) \simeq A \otimes F(X), \\ 1\boxtimes X\simeq X,\\ (A\otimes B)\boxtimes X\simeq A\boxtimes (B\boxtimes X).$$ Satisfying some expected coherence diagrams. Is this condition sufficient? What about the nice tensor categories mentioned above?
(For a co-module $X$ over a co-algebra $A$, with the structure map $\rho: X \to X\otimes A$,the box product $Y\boxtimes X$ is defined to be $Y\otimes X$ with co-action $$Y\otimes \rho:Y\otimes X\to Y\otimes X\otimes A.$$)