2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78180The basic method of rewriting for words in a free monoid given a monoid presentation is extended to rewriting for paths in a free category given a `Kan extension presentation'. This is related to work of Carmody-Walters on the Todd-Coxeter procedure for Kan extensions, but allows for the output data to be infinite, described by a language. The result also allows rewrite methods to be applied in a greater range of situations and examples, in terms of induced actions of monoids, categories, groups or groupoids.31 pages, LaTeX2e, (submitted to JSC)Combinatorics68Q42 18A40 68Q40text