nLab natural isomorphism



Category theory

Equality and Equivalence



A natural isomorphism η:FG\eta\colon F \Rightarrow G between two functors FF and GG

F C η D G \array{ & \nearrow \searrow^{F} \\ C &{}^{\simeq}\Downarrow^\eta& D \\ & \searrow \nearrow_{G} }

is equivalently

In this case, we say that FF and GG are naturally isomorphic. (Synonym: FF and GG are isomorphic functors; the naturality is understood when one says that two functors are isomorphic.)

Notably, especially in expositions and lectures, despite the third bullet point above, one does not need to define the concept of a functor category in order to define isomorphic functors. Authors sometimes make cautionary remarks about the category of all functors from CC to DD (cf. e.g. Auslander (1971, p.9))

If you want to speak of ‘the’ functor satisfying certain conditions, then it should be unique up to unique natural isomorphism.

A natural isomorphism from a functor to itself is also called a natural automorphism.

Some basic uses of isomorphic functors

Defining the concept of equivalence of categories

A fundamental use of the concept of isomorphic functors is the usual definition of equivalent categories which involves functors isomorphic to identity functors.

Re-defining isomorphism of objects in terms of isomorphism of functors

The Yoneda lemma implies that in any category 𝒞\mathcal{C}, and for any objects OO and OO' of 𝒞\mathcal{C}, the following are equivalent:


  • M. Auslander?, The representation dimension of artin algebras.

    Queen Mary College Mathematics Notes (1971) Republished in: Selected works of Maurice Auslander. American Mathematical Society (1999)

Last revised on December 7, 2023 at 03:50:08. See the history of this page for a list of all contributions to it.