definition. natural isomorphism [kostecki2011introduction, 4.2] [tt-001H]

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A natural transformation \(\sigma : \mathscr {F} \to \mathscr {G}\) between functors \(\mathscr {F}: {\cal C} \to {\cal D}\) and \(\mathscr {G} : {\cal C} \to {\cal D}\) is called a natural isomorphism or a natural equivalence, denoted \(\sigma : \mathscr {F} \cong \mathscr {G}\), if each component \(\sigma _X : \mathscr {F}(X) \to \mathscr {G}(X)\) is an isomorphism in \({\cal D}\), i.e. \(\mathscr {F}(X) \underset {\sigma _X}{\cong } \mathscr {G}(X)\).

We call \(\mathscr {F}\) and \(\mathscr {G}\) naturally isomorphic to each other.

We also say that \(\mathscr {F}(X) \cong \mathscr {G}(X)\) naturally in \(X\) [leinster2016basic, 1.3.12].

Diagramatically,