I added a sentence to explain the ’doctrine’ part of ’doctrinal adjunction’.

]]>Thanks, I see now where I made a mistake in parsing.

Maybe I’ll find some time to re-arrange the notation of the entry a little. On my system at least, it is somewhat hard to decypher, with different kinds of $n$-morphisms all in the same font and with the difference between small twiddles and small bars hard to make out.

Generally, for better reading experience, I found that

```
\overline{}
```

beats

```
\bar{ }
```

and it is good to type

```
\widetilde
```

even for single symbols to be decorated. But in the entry at hand, maybe one should find altogether different decoration, for readability.

]]>Actually, it would be enough to stop at “have an inverse.” The fact that this makes $(f,\bar{f})$ into a lax and a strong $T$-morphism then follows automatically. But $\bar{f}$ is a 2-morphism here too, since it is an inverse of the 2-morphism $\tilde{f}$.

]]>Is the statement of prop. 2.2 at *doctrinal adjunction* as intended:

For the unit and counit of the adjunction $f \dashv u$ to be $T$-transformations, and hence for the adjunction to live in $T$-$Alg_l$, it is necessary and sufficient that $\tilde f$ have an inverse $\bar f$ that makes $(f,\bar f)$ into a lax $T$-morphism, and hence $(f,\bar f)$ into a strong $T$-morphism.

?

The last line repeats the symbls $(f,\bar f)$. If this is what is really meant, it would be more clear to write “into a lax $T$-morphism, which is then necessarily a strong $T$-morphism”.

And in the lines before the proposition, symbols “$\bar f$” refer to 2-morphisms, while here they refer to (inverses of) 1-morphisms. I am not sure if I am parsing this correctly.

]]>Thanks! I added some remarks about the way I prefer to think of doctrinal adjunction in terms of double categories. This motivated me to finally create companion pair and conjunction.

]]>Created doctrinal adjunction. The page could probably use some examples and/or fleshing out.

]]>