nLab
natural isomorphism

Definition

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

$$\begin{array}{cc}& \nearrow {\searrow }^F\\ C& {}^{\simeq }{\Downarrow }^{\eta }& D\\ & \searrow {\nearrow }_G\end{array}$$\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 $c\in \mathrm{Obj}(C)$ 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.

Related concepts

• natural isomorphism

• natural equivalence