i once heard michael johnson talk about "posets with all pullbacks", and i helpfully pointed out to him that it's silly to talk about pullbacks in posets because a pullback in a poset is just a product...

(as it turned out i was the one being a bit silly...)

it might be interesting to categorify this phenomenon now in a certain way.... perhaps there's some concept of "category with homotopy-limits of all diagrams on 2-connected diagram schemes" (or something like that) which could be misconstrued as silly in an analogous way... ?and which might even be relevant to some things that we've been thinking about recently?