Added an adjoint functor theorem for cocomplete categories.

]]>Change notation in the statement of the theorem to match its proof (the functor is $R:C\to D$ instead of $G:D\to C$).

]]>Clarified the language in another relevant spot (where a counterexample was given).

]]>Clarified some language in the statements that characterize adjoints between locally presentable categories, in response to a comment made by user Hurkyl in another thread (here).

]]>Thanks.

]]>I went ahead and made some changes per your comment. See if that looks better. (I think I’d try a different explanation if I were writing this – or writing this today in case I was the one who wrote that then! – but never mind.)

]]>I fixed a trivial typo in adjoint functor theorem but left wondering about this:

… the limit

$L c := \lim_{c\to R d} d$over the comma category $c/R$ (whose objects are pairs $(d,f:c\to R d)$ and whose morphisms are arrows $d\to d'$ in $D$ making the obvious triangle commute in $C$) of the projection functor

$L c = \lim_{\leftarrow} (c/R \to D ) \,.$

I don’t really understand this (and while I could figure it out, it’s probably not good to make readers do so). At first it sounds like someone is saying “the limit $L c$ over the comma category of the projection functor $L c$”, which would be circular. But it must be that both formulas are intended as synonymous definitions of $L c$. At that point one is left wondering why one has a backwards arrow under it and the other does not. I guess old-fashioned people prefer writing limits with backwards arrows under them, so someone is trying to cater to all tastes? I think it’s better in this website to use $lim$ and $colim$ for limit and colimit.

I could probably guess how to fix this, but I won’t since I might screw something up.

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