$newcommandVect{mathrm{Vect}}newcommandHom{mathrm{Hom}}$Let $A$ be a (small, whatever else we need) $k$-linear category for a field $k$. Let $M$ be an $A$-module, i.e., a functor $Mcolon A to Vect$, and $N$ be an $A^text{op}$-module. $AWe can form the tensor product

$$M otimes_A N = int^{a in A} Ma otimes_k Na,$$

which lies in $Vect$. This should be enough to define $mathrm{Tor}^A_i(M, N)$ by the long exact sequence. I have questions about this functor:

- Is it balanced? I assume yes. Does this already follow from the Embedding Theorem and balancedness for modules?
- Is it computable in terms of free modules? What is a free module, even? I know about the free-forgetful adjunction, but is there a more hands-on description of free modules? Is $Hom_A(a,-)$ for $a in A$ a free module? Are all free modules direct sums of this?
- Are projective modules flat?

I’d guess that all answers to these questions are affirmative. Where can I find more about that?