# 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. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/450-category-theory/10-morphisms/inverse-morphism.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.