  We provide, among other things: (i) a Bousfield--Kan formula for colimits in
$\infty$-categories (generalizing the 1-categorical formula for a colimit as a
coequalizer of maps between coproducts); (ii) $\infty$-categorical
generalizations of Barwick--Kan's Theorem B$_n$ and Dwyer--Kan--Smith's Theorem
C$_n$ (regarding homotopy pullbacks in the Thomason model structure, which
themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of
the simultaneous and interwoven functoriality of colimits (or dually, of
limits) for natural transformations and for pullback along maps of diagram
$\infty$-categories.
