Eklund et al. (2002) present a graphical technique aimed at simplifying the verification of various category-theoretic constructions, notably the composition of monads. In this note we take a different approach involving string rewriting. We show that a given tuple $(T,\mu,\eta)$ is a monad if and only if $T$ is a terminal object in a certain category of strings and rewrite rules, and that this fact can be established by proving confluence of the rewrite system. We illustrate the technique on the monad composition problem. We also give a characterization of adjunctions in terms of rewrite categories.

• 18A23 - Natural morphisms, dinatural morphisms
• 18A40 - Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
• 18C15 - Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
• 68Q42 - Grammars and rewriting systems