Assume $(C,\otimes,e)$ to be a strict abelian tensor category with additive tensor functor and $A$ an algebra object in $C$. Then we're able to define the category of $A$-right modules $C_A$ and the category of $A$-left modules $_AC$.
For an $A$-right module $(M,\triangleleft)$ and an $A$-left module $(N,\triangleright)$ we define the tensor product $M\otimes_A N$ to be given by the cokernel of $\triangleleft\otimes id- id \otimes\triangleright:M\otimes A \otimes N \rightarrow M \otimes N$. This definition is indeed functorial, i.e. we have $\otimes_A: C_A \times _AC \rightarrow C$. Now assume $C_A$ to be abelian and fix an $A$-right module $N$. Then we obtain an functor $-\otimes_A N:C_A \rightarrow C $.
If the tensor product $\otimes$ of $C$ is exact, one can show, that $-\otimes_A N$ is exact by using the snake lemma. However I fail to see, whether this holds in a more general setting.
Is there some sort of adjointness one could use to generealise this statement?
(See also Exercise 7.8.23 in the book "Tensor Categories" by P. Etingof, S. Gelaki, D. Nikshych and V.Ostrik).