MATH.16:10 - Architectural Rationale

Organizing the choice around maps makes the required operations explicit before committing to an encoding. Including uniqueness expresses when the supplied data determine the whole result. It also gives reusable arguments for comparing maps and comparing realizations.

A direct representation remains economical for a single familiar calculation. The universal-property method earns its additional abstraction when several constructions are plausible, a representation must be changed, or later maps and arguments need to be derived. The product, pullback and coproduct cases expose different requirements. The prepared-function case shows how to formulate another requirement by varying its settings and behavior while keeping the input and output sets fixed.

Detailed quotient, generator-extension and structure-transport methods remain separately usable. They supply constructions and proofs after this method has identified the needed property. This separation permits a broader choice method without compressing those techniques into unexplained instructions.