Isomorphism of categories

In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1_D (the identity functor on D) and GF = 1_C. This means that both the objects and the morphisms of C and D stand in a one-to-one correspondence to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.

Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of equivalence of categories; roughly speaking, for an equivalence of categories we don't require that ${\displaystyle FG}$ be equal to ${\displaystyle 1_{D}}$, but only naturally isomorphic to ${\displaystyle 1_{D}}$, and likewise that ${\displaystyle GF}$ be naturally isomorphic to ${\displaystyle 1_{C}}$.

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