[[Natural transformation]]
# Natural isomorphism

A **natural isomorphism** is an isomorphism in a [[functor category]], #m/def/cat 
i.e. a [[natural transformation]] $\eta \in \cat D^{\cat C}(F, G)$
such that $\eta_{X} : FX \to GX$ is an [[Morphism|isomorphism]] for all $X \in \cat C$.
If such an isomorphism exists we write $F \simeq G$.

The idea was first proposed in [[@eilenbergGeneralTheoryNatural1945|A general theory of natural equivalences]], which is also the originating paper of category theory.

See [[Equivalence of categories]]

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