Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care most about the situation where $\mathcal C$ is *finite $\mathbb C$-linear* in the sense of arXiv:1406.4204 (in which case the monoidal structure is closed iff it is right exact in each variable). Note that, unlike in that paper, I specifically care about monoidal structures which are not rigid. An example of the category I have in mind is the category $\mathrm{Mod}^f_A$ of finite-dimensional modules for any finite-dimensional commutative algebra $A$ (with $\otimes = \otimes_A$).

Recall that an object $P \in \mathcal C$ is *projective* if $\hom(P,-) : \mathcal C \to \mathrm{AbGp}$ is right exact. (It is already left exact.) Note that this has nothing to do with the monoidal structure.

An object $F \in \mathcal C$ is *flat* if $F \otimes : \mathcal C \to \mathcal C$ is left exact. (It is already right exact.) Note that this has everything to do with the monoidal structure.

Are projective objects necessarily flat?

Of course, in $\mathrm{Mod}_A^f$ they are. The other examples I usually use of non-rigid monoidal categories are the representation theories of non-Hopf bialgebras, but there every object is flat.