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The symbol $\cong$ denotes an isomorphism in the category of sets, which means it is simply a bijection between the two hom-sets. Because hom-sets are sets, their isomorphisms are bijective functions.

Conditions: The compared objects are hom-sets; The ambient category for the isomorphism is the category of sets

The symbol $\cong$ denotes an isomorphism in the category of sets, which means it is simply a bijection between the two hom-sets. Because hom-sets are sets, their isomorphisms are bijective functions.

Conditions: The compared objects are hom-sets; The ambient category for the isomorphism is the category of sets