I know that a project module is always flat, deduced form the properties and abundance of free modules. I'm trying to figure out how essential role the free modules play in this result. So I'd like to examine if this result hold for more general cases. More precisely:
Given a closed symmetric monoidal abelian category $(\mathcal A, +, I, \otimes)$, we can plainly generalize the notion of projective object and flat object to $\mathcal A$. Is a projective object of $\mathcal A$ always flat?
EDIT: the question has been already asked and answered at Mathoverflow.