MATH.16:4.3 - Construct an object and establish the property

In sets, the product is the set of ordered pairs A x B. Use the coordinate projections and define <f,g>(x)=(f(x),g(x)). Both projection equations follow by reading the corresponding coordinate. Any function with those projections must return that same pair at every x, which proves uniqueness.

For the pullback, form the subset:

Q={(a,b) in A x B | s(a)=t(b)}.

The same pairing formula lands in Q exactly when the compatibility equation holds. Reading the coordinates again proves uniqueness. Q may be empty. The empty set is still a valid pullback in sets; it says no pair meets the stated compatibility requirement.

For a coproduct of sets, distinguish the two input branches with tags 0 and 1:

S=({0} x A) union ({1} x B).

The injections are iA(a)=(0,a) and iB(b)=(1,b). Define [f,g](0,a)=f(a) and [f,g](1,b)=g(b). Every element has one of these forms, which establishes existence and forces the rule uniquely. Even when A and B are the same set, the tags keep the two branches distinct. For A=B={0}, handlers f(0)=0 and g(0)=1 therefore extend to a function taking (0,0) to 0 and (1,0) to 1.

Additional structure – groups and topology. Establish that the object and maps have the chosen structure. For groups, define product multiplication componentwise. If f and g preserve multiplication, then their paired map does too, because each coordinate does. For a pullback of group homomorphisms, the compatibility equation is preserved by multiplication and inverses. A construction with continuous maps requires the appropriate topology and continuity arguments.

Use a known applicable construction when available. For quotient and generator questions, MATH.2 and MATH.5 provide the detailed operations. If the required object cannot yet be constructed, identify the unresolved existence or construction question. An existence theorem may supply a result for reasoning while leaving an effective way to obtain its elements for further work.