| context | $F$ in ${\bf D}\longrightarrow{\bf C}$ |
| context | $G$ in ${\bf C}\longrightarrow{\bf D}$ |
| definiendum | $\langle\alpha,\beta\rangle$ in $F\simeq G$ |
| inclusion | $\alpha, \beta$ … my nice nats $\left(F,G\right)$ |
| inclusion | $\alpha,\beta$ … natural isomorphism |
$\alpha$ in $FG\cong Id_{\bf C}$
$\beta$ in $Id_{\bf D}\cong GF$.
Note the two different symbols $\cong$ and $\simeq$. The first is about equivalences, the second about invertible gadgets.
This is like equivalence of categories, except the natural transformations are not just required to exist but must be concretely specified. As such, this is a subset of my nice nats.
Wikipedia: Equivalence of categories