Universal Property
Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i1 : X → P and i2 : Y → P for which the following diagram commutes:
Moreover, the pushout (P, i1, i2) must be universal with respect to this diagram. That is, for any other such set (Q, j1, j2) for which the following diagram commutes, there must exist a unique u : P → Q also making the diagram commute:
As with all universal constructions, the pushout, if it exists, is unique up to a unique isomorphism.
Read more about this topic: Pushout (category Theory)
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