  Adhesive and quasiadhesive categories provide a general framework for the
study of algebraic graph rewriting systems. In a quasiadhesive category any two
regular subobjects have a join which is again a regular subobject. Vice versa,
if regular monos are adhesive, then the existence of a regular join for any
pair of regular subobjects entails quasiadhesivity. It is also known
(quasi)adhesive categories can be embedded in a Grothendieck topos via a
functor preserving pullbacks and pushouts along (regular) monomorphisms. In
this paper we extend these results to $\mathcal{M}, \mathcal{N}$-adhesive
categories, a concept recently introduced to generalize the notion of
(quasi)adhesivity. We introduce the notion of $\mathcal{N}$-adhesive morphism,
which allows us to express $\mathcal{M}, \mathcal{N}$-adhesivity as a condition
on the subobjects's posets. Moreover, $\mathcal{N}$-adhesive morphisms allows
us to show how an $\mathcal{M},\mathcal{N}$-adhesive category can be embedded
into a Grothendieck topos, preserving pullbacks and $\mathcal{M},
\mathcal{N}$-pushouts.
