  A foundational theory of compositional categorical rewriting theory is
presented, based on a collection of fibration-like properties that collectively
induce and intrinsically structure the large collection of lemmata used in the
proofs of theorems such as concurrency and associativity. The resulting highly
generic proofs of these theorems are given. It is noteworthy that the proof of
the concurrency theorem takes only a few lines and, while that of associativity
remains somewhat longer, it would be unreadably long if written directly in
terms of the basic lemmata. In essence, our framework improves the readability
and ease of comprehension of these proofs by exposing latent modularity. A
curated list of known instances of our framework is used to conclude the paper
with a detailed discussion of the conditions under which the Double Pushout and
Sesqui-Pushout semantics of graph transformation are compositional.
