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I suppose we ought also to have a page on the double category of adjunctions that figures in the mate correspondence.
I see at mate it speaks about the double category, $Adj(K)$. But this is the notation I just used for the 2-category of adjunctions in $K$.
Yes, well, in a latex paper I would use two different fonts for the “$Adj$”.
Re #5, can’t we have two fonts here? Which should they be?
I would probably write $\mathcal{A}\mathit{dj}(K)$ for the 2-category and $\mathbb{A}\mathsf{dj}(K)$ for the double category. That doesn’t work in a page title, though.
[I wish I could have deleted this comment.]
The inclusion of $Mnd$, the free monad, in $Adj$ induces a 2-functor from the 2-category of adjunctions in $K$ to the 2-category of monads in $K$.
This doesn’t make sense to me. $Adj(K)$ is not the functor 2-category $[Adj,K]$ – as it says earlier on the page, the objects of $Adj(K)$ are the objects of $K$ while its morphisms are the functors $Adj\to K$ – so I don’t see any “precomposition” functor going on.
Have I garbled the end of The free adjunction?
Yes, their “2-category of adjunctions” is by definition $[Adj,K]$, not the 2-category we’re calling $Adj(K)$ on this page. Perhaps this page should mention both, since this confusion seems likely to be common.
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