Added a better example which Rune Haugseng explained to me.

]]>Thanks! “Roughly equivalent” notions probably appear in different guises elsewhere too, of course.

]]>I know these from Schumacher & Street’s paper on Some parametrized categorical concepts as “cartesian cones”. Also, in the case of diagrams indexed by discrete categories, I think these are the “initial lifts” of “structured sources” from The Joy of Cats.

]]>Added some basic examples from HTT. There doesn’t seem to be a page for the corresopnding 1-categorical notion. This notion is used pretty heavily in $\infty$-category theory, but it’s not so familiar from 1-category theory. But I’d have to think a 1-categorical treatment exists *somewhere*, right?