MATH.16:4.4 - Derive the next map, comparison or operation

Use the universal property to build the map the question needs. For a product, give the two component maps. For a pullback, also establish their agreement in C. For a coproduct, give a processing rule for each input kind.

The property also supplies equality tests. Two maps into a product or pullback are equal when both their projections agree. Two maps out of a coproduct are equal when composing each with iA gives equal maps and composing each with iB gives equal maps. This lets you compare an entire map through its required parts.

It can compare different constructions of the same object. If P and P’ satisfy the same product property for A and B, the projections of each determine a map to the other. Composing these maps preserves both projections. The identity does too, so uniqueness makes the composites identities. Thus the two constructions are isomorphic by the maps that preserve their projections. MATH.7 develops the transport of further structure when a usable bijection has been obtained.

A new operation on the components needs a compatibility test. Here take Q to be the pullback of sets and u:A -> A, v:B -> B to be functions proposing updates. They induce a function (a,b) -> (u(a),v(b)) on Q exactly when:

s(u(a))=t(v(b)) whenever s(a)=t(b).

The original agreement does not settle the changed one. A failed pair identifies which update or agreement condition must change. This gives a way to work on operations themselves while keeping their required uses visible.

Additional structure – groups. When the update must also be a homomorphism, establish preservation of the group operation. Agreement of the updated components alone establishes only a function on the set of compatible pairs.