equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
The identity functor on a category $C$ is the functor $id_C: C \to C$ that maps each object and morphism of $C$ to itself. The identity functors are the identities for composition of functors in Cat.
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