  We introduce the dual notions of $\mathcal{E}(\mathcal{X},M,\mathcal{Y})$ and
$\mathcal{M}(\mathcal{X},M,\mathcal{Y})$, and investigate when they have enough
injective objects or projective objects, when they are resolving or
co-resolving, and when they are Frobenius categories. In the case that they are
Frobenius categories, we establish a recollement of their stable categories.
Finally some applications to comma categories are given.
