Amnestic functor

In the mathematical field of category theory, an amnestic functor F : A → B is a functor for which an A-isomorphism ƒ is an identity whenever Fƒ is an identity.

An example of a functor which is not amnestic is the forgetful functor Met_c→Top from the category of metric spaces with continuous functions for morphisms to the category of topological spaces. If ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$ are equivalent metrics on a space ${\displaystyle X}$ then ${\displaystyle \operatorname {id} \colon (X,d_{1})\to (X,d_{2})}$ is an isomorphism that covers the identity, but is not an identity morphism (its domain and codomain are not equal).