===== My nice nats =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: $F$ in ${\bf D}\longrightarrow{\bf C}$ |
| @#55CCEE: context     | @#55CCEE: $G$ in ${\bf C}\longrightarrow{\bf D}$ |
| @#FFBB00: definiendum | @#FFBB00: $\langle\alpha,\beta\rangle$ in it |
| @#AAFFAA: inclusion   | @#AAFFAA: $\alpha:FG\xrightarrow{\bullet}1_{\bf C}$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\beta:1_{\bf D}\xrightarrow{\bullet}GF$ |

==== 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.

==== Parents ====
=== Context ===
[[Functor]]
=== Requirements ===
[[Natural transformation]], [[Identity functor]]