category theory

# Contents

## Definition

A natural isomorphism $\eta\colon F \Rightarrow G$ between two functors $F$ and $G$

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

is equivalently

• a natural transformation with a two-sided inverse;

• a natural transformation each of whose components $\eta_c : F(c) \to G(c)$ for all $c \in Obj(C)$ is an isomorphism in $D$;

• an isomorphism in the functor category $[C,D]$.

In this case, we say that $F$ and $G$ are naturally isomorphic.

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.

Revised on January 17, 2017 12:40:58 by Toby Bartels (108.167.41.14)