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We introduce a category-theoreticabstraction of a syntax with auxiliaryfunctions, called an admissiblemonad morphism. Relying on an abstract form ofstructural recursion,we then design generic tools to construct admissible monadmorphismsfrom basic data. These tools automate ubiquitous standard patternslike(1) defining auxiliary functions in successive, potentiallydependent layers,and (2) proving properties of auxiliary functions byinduction on syntax. Wecover significant examples from theliterature, including the standardlambda-calculus withcapture-avoiding substitution, a lambda-calculus withbindingevaluation contexts, the lambda-mu-calculus with named substitution,andthe differential lambda-calculus.

A unified treatment of structural definitions on syntax for  capture-avoiding substitution, context application, named substitution,  partial differentiation, and so on