Free and co-free constructions for Hopf categories

We show that under mild conditions on the monoidal base category $\mathcal{V}$, the category $\mathsf{V}\mathsf{H}\mathsf{o}\mathsf{p}\mathsf{f}$ of Hopf $\mathcal{V}$-categories is locally presentable and deduce the existence of free and cofree Hopf categories. We also provide an explicit description of the free and cofree Hopf categories over a semi-Hopf category. One of the conditions on the base category $\mathcal{V}$, states that endofunctors obtained by tensoring with a fixed object preserve jointly monic families, which leads us to the notion of ``very flat monoidal product’', which we investigate in particular for module categories.