Natural isomorphism

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Definition

Symbol-free definition

A natural isomorphism between functors is a natural transformation between the functors that has a two-sided inverse which is also a natural transformation.

Definition with symbols

Suppose ${\mathcal {C}},{\mathcal {D}}$ are categories and ${\mathcal {F}},{\mathcal {G}}:{\mathcal {C}}\to {\mathcal {D}}$ are functors. A natural isomorphism is a natural transformation $\eta :{\mathcal {F}}\to {\mathcal {G}}$ such that there exists a natural transformation $\nu :{\mathcal {G}}\to {\mathcal {F}}$ such that the composites $\eta \circ \nu$ and $\nu \circ \eta$ are both identity natural transformations. Specifically, $\eta \circ \nu$ is the identity transformation from ${\mathcal {G}}$ to itself, and $\nu \circ \eta$ is the identity transformation from ${\mathcal {F}}$ to itself.