Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules $\mathcal{F}$such that direct sums of $\mathcal{F}$-covers ( $\mathcal{F}$-envelopes) are $\mathcal{F}$-covers ( $\mathcal{F}$-envelopes), $\mathcal{F}$-covering ( $\mathcal{F}$-enveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of $\mathcal{F}$-covers are $\mathcal{F}$-covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.