MATH.16:4.2 - State the universal property before choosing an encoding

Turn the use into a requirement by comparing the proposed object with arbitrary permitted ways of supplying or processing its data:

1. Fix the objects already given by the question, together with their maps and laws. They remain the same throughout this comparison.
2. Introduce a variable object X for a possible source of data, or Z for a possible destination. Describe the maps and equations that make it an allowed instance of the needed use. Let it vary over every permitted instance, not just one sample. For example, a source of two components supplies functions from X to each fixed component set.
3. Seek an object P with the maps through which it will be used. Ask how an allowed instance should relate to P: should its data assemble into P, or should processing extend from P to a destination? Draw that comparison map in the corresponding direction.
4. Combine the comparison map with P’s use maps in the order their domains require, and equate the resulting routes with the original maps. Require a comparison map for every allowed instance, and uniqueness when the supplied data should determine it completely.

This specifies the behavior a construction must realize. If the intended use leaves the maps or agreements undecided, return to that particular choice before asking for a universal object. The prepared-function case in :5.4 follows these steps for a use beyond pairing or tagging.

Suppose the question requires two recoverable components in A and B, and those components must determine the entire result. A and B are fixed; a trial source X supplies maps to both. Seek an object P with projections pA:P -> A and pB:P -> B.

For every allowed object X and maps f:X -> A, g:X -> B, require a unique map <f,g>:X -> P such that:

pA composed with <f,g> = f;

pB composed with <f,g> = g.

These equations say that forming the combined result and reading either component returns the supplied component. Uniqueness says there is no further choice in that combination. Any map h:X -> P is consequently recovered from its components:

<pA composed with h, pB composed with h> = h.

An object with these projections and property is a product of A and B. Its specification describes both how to construct a result and how to use it.

Now suppose the components must agree through maps s:A -> C and t:B -> C. Seek projections that satisfy:

s composed with pA = t composed with pB.

Require a unique combined map only for f and g satisfying s composed with f = t composed with g. This is a pullback of s and t: it combines precisely the data compatible under that equation.

To choose an object for either kind of input, reverse the mapping question. Seek maps iA:A -> S and iB:B -> S that place the inputs in the new object. For maps f:A -> Z and g:B -> Z, require a unique map [f,g]:S -> Z with [f,g] composed with iA = f and [f,g] composed with iB = g. This is a coproduct. In sets, a tag can preserve which input was supplied, so the processing rule can select the correct branch.

The comparison points into the product or pullback and out of the coproduct. Its direction follows the needed operation: assemble supplied components or extend supplied processing rules.