  Coherence theorems for covariant structures carried by a category have
traditionally relied on the underlying term rewriting system of the structure
being terminating and confluent. While this holds in a variety of cases, it is
not a feature that is inherent to the coherence problem itself. This is
demonstrated by the theory of iterated monoidal categories, which model
iterated loop spaces and have a coherence theorem but fail to be confluent. We
develop a framework for expressing coherence problems in terms of term
rewriting systems equipped with a two dimensional congruence. Within this
framework we provide general solutions to two related coherence theorems:
Determining whether there is a decision procedure for the commutativity of
diagrams in the resulting structure and determining sufficient conditions
ensuring that ``all diagrams commute''. The resulting coherence theorems rely
on neither the termination nor the confluence of the underlying rewriting
system. We apply the theory to iterated monoidal categories and obtain a new,
conceptual proof of their coherence theorem.
