In many categories arising in the theory of algebras or modules, projective objects are flat. For example each projective module over a ring with identity is flat. However, there are categories where this principle is violated: Lewis has shown that the category of Mackey functors for the orthogonal group $O(n)$ has non-flat projectives.

**Q1:** Is there a classification of symmetric monoidal abelian categories $\mathcal{A}$ where all projectives are flat ? BTW: Do such categories have a particular name ?

If a classification is too strong to establish, I would be interested in

**Q2:** Are there criteria that ensure that projectives are flat ?

An example for Q2 is, when $\mathcal{A}$ has direct sums and the unit is a progenerator of $\mathcal{A}$ (this is just the categorification of the module case mentioned above). Note that this criteria doesn't work for instance in the category of $RG$-modules ($R$ commutative ring, $G$ a group) where the tensor product is given by $M \otimes_R N$ with diagonal $G$-operation.