MATH.16:4.5 - Use the result and return to a changed requirement

Return the construction with the maps and conditions needed by its consumer. A name such as “product” helps recognition; the projections, formation rule and applicable laws let the receiver use it.

For a mathematical question, stop with the required map, equality, usable construction or demonstrated obstruction. For computation, obtain the procedure and resources needed for the chosen representation through C.29.2. A finite pullback can be enumerated by testing pairs, while a large or infinite one needs an appropriate computational method.

For an application to another subject, C.29 supplies the correspondence that gives the mathematical objects and equations their subject meaning. In particular, a compatibility equation must represent the actual agreement needed by the work. A pairing of functions on one input describes a different operation from two executions that modify a shared input. For the latter use, first specify which values each step reads and changes, and which intervening steps are permitted. Use that account to decide whether a function construction represents the work; the pairing alone leaves those interactions unspecified.

When the question changes, return to the affected mapping requirement. Adding agreement can turn a product question into a pullback question. Needing to accept either input can call for a coproduct. Needing only selected answers can call for a quotient. Retain a useful earlier construction while its earlier question remains current.