natural isomorphism
1. Definition
Let \(\mathcal{C}, \mathcal{D}\) be categories and \(\mathcal{F}, \mathcal{G}: \mathcal{C} \rightarrow \mathcal{D}\) be functors.
A natural isomorphism is a natural transformation \(\eta: \mathcal{F} \Rightarrow \mathcal{G}\) with a two-sided inverse \(\eta^{-1}: \mathcal{G} \Rightarrow \mathcal{F}\).
i.e. for the composition it holds
\begin{align*} \eta \circ \eta^{-1} =& \mathrm{id}_{\mathcal{G}} \\ \eta^{-1} \circ \eta =& \mathrm{id}_{\mathcal{F}} \end{align*}