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this is a bare little section, to be !include
-ed as a Properties-subsection at slice category and adjoint (∞,1)-functor (where two copies of this same section used to be all along) and also at slice (∞,1)-category and adjoint functor (where, for completeness, the same should be recorded, too, but wasn’t until now)
It would also be good to expand a little here, for instance by adding a pointer to a 1-category textbook account (this is probably in Borceux, but I haven’t checked yet), or, of course, by adding some indication of the proof.
I have added the proof of the slice adjunction
$\mathcal{D}_{/L(c)} \underoverset {\underset{\;\;R_{/c}\;\;}{\longrightarrow}} {\overset{\;\;L_{/c}\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/c}$in 1-category theory – and in tikzcd diagrams.
The typesetting looks quite neat in my local pdf rendering. On the $n$Lab it currently comes out not quite as neat since:
(1) the font size of the tikzcd diagrams does not match that of the ambient text,
(2) tikzcd diagrams can’t be put inline here, or even just next to each others in a line.
But I guess it’s okay.
added statement and proof also of the other slicing
$\mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)}$What’s a canonical reference for this? This ought to be in Borceux somewhere, but I haven’t looked for it yet.
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