Inverse morphism

A morphism f ∶ a → b in a category 𝒞 is invertable if there is a morphism f⁻¹ ∶ b → a such that f⁻¹ ∘ f = 1ᴀ and f ∘ f⁻¹ = 1ʙ.

An invertable arrow is an isomorphism.

In some fields of category theory, besides the usual categorical axioms, one can also introduce the restriction that all arrows must have inverses, i.e. that all arrows be isomorphic, and then investigate things under that set of constraints. Also, some other field of CT might relax the axioms, for example, by allowing weaker version of associativity of composition, where different compositions almost amount to the same thing.