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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 = 1D and GF = 1C. 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 F G {\displaystyle FG} be equal to 1 D {\displaystyle 1_{D}} , but only naturally isomorphic to 1 D {\displaystyle 1_{D}} , and likewise that G F {\displaystyle GF} be naturally isomorphic to 1 C {\displaystyle 1_{C}}.