## My equivalence of categories

### Collection

 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

### Discussion

#### Elaboration

$\alpha$ 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.

#### Idea

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.