## My nice nats

### Collection

context | $F$ in ${\bf D}\longrightarrow{\bf C}$ |

context | $G$ in ${\bf C}\longrightarrow{\bf D}$ |

definiendum | $\langle\alpha,\beta\rangle$ in it |

inclusion | $\alpha:FG\xrightarrow{\bullet}1_{\bf C}$ |

inclusion | $\beta:1_{\bf D}\xrightarrow{\bullet}GF$ |

### Discussion

That silly name … I made it up.

The natural transformation $\beta:1_{\bf D}\xrightarrow{\bullet}GF$ squeezes every set $X\in {\bf D}$ into a set $GFX\in {\bf D}$ (although this need not be surjective or injective). The natural transformation $\alpha:FG\xrightarrow{\bullet}1_{\bf C}$ squeezes all sets $FGX$ in the image of $FG$ back into $X$. The latter operation gets rid of lots $FG$'s without changing the structural properties of ${\bf C}$.

The point is that my equivalence of categories and Counit-unit adjunction are two different important special cases of nice nats. In the former case, the two nats actually shift the whole content of a category internally. In the latter case, the two nats end up defining the shifting operations of a monad.

#### Theorems

Only when the nats are isomorphisms (as in my equivalence of categories) is $F$ fully faithful and dense.