Natural isomorphism

Let $F,G:\mathcal C\to\mathcal D$ be functors .

Definition

A natural isomorphism $\eta:F\Rightarrow G$ is a natural transformation such that for every object $X\in\mathcal C$, the component

is an isomorphism in $\mathcal D$.

Equivalently, $\eta$ is a natural isomorphism iff there exists a natural transformation $\eta^{-1}:G\Rightarrow F$ such that $\eta^{-1}_X=(\eta_X)^{-1}$ for all $X$ (and hence $\eta^{-1}\circ \eta=\mathrm{id}_F$, $\eta\circ \eta^{-1}=\mathrm{id}_G$).

Examples

1. Double dual on finite-dimensional vector spaces.
   In $\mathbf{FinVect}_k$, the canonical maps $V\to V^{\ast\ast}$ assemble to a natural transformation $\mathrm{Id}\Rightarrow (-)^{\ast\ast}$, and each component is an isomorphism. Hence

   $$\mathrm{Id}\;\cong\;(-)^{\ast\ast}$$

   naturally on $\mathbf{FinVect}_k$.

2. Swap of factors for products.
   In any category with binary products , there is a natural isomorphism

   $$\sigma_{A,B}:A\times B \xrightarrow{\cong} B\times A$$

   characterized by $\pi_1\circ \sigma_{A,B}=\pi_2$ and $\pi_2\circ \sigma_{A,B}=\pi_1$.
   In $\mathbf{Set}$ it is the function $(a,b)\mapsto(b,a)$; in $\mathbf{Grp}$ it is the group homomorphism $(g,h)\mapsto(h,g)$.

3. Swap of summands for coproducts.
   Dually, in any category with binary coproducts , there is a natural isomorphism

   $$A\sqcup B \xrightarrow{\cong} B\sqcup A$$

   induced by the universal property of the coproduct.
   In $\mathbf{Set}$ this is the obvious bijection between disjoint unions; in $\mathbf{Ab}$ it is the canonical isomorphism $A\oplus B\cong B\oplus A$.