  In this contribution we investigate several extensions of the powerset that
comprise arbitrarily nested subsets, and call them superpower set. This allows
the definition of graphs with possibly infinitely nested nodes. additionally we
define edges that are incident to edges. Since we use coalgebraic constructions
we refer to these graphs as corecursive graphs. The superpower set functors are
examined and then used for the definition of $\mathcal{M}$-adhesive categories
which are the basic categories for $\mathcal{M}$-adhesive transformation
systems. So, we additionally show that coalgebras $\mathbf{Sets}_F$ are
$\mathcal{M}$-adhesive categories provided the functor $F:\mathbf{Sets}_F \to
\mathbf{Sets}_F$ preserves pullbacks along monomorphisms.
