5:18 PM
@OmarAntolín-Camarena so take an arbitrary sequence of spaces Y_0 -> Y_1 -> Y_2 -> .... define your poset P to be ℕ x ℤ with lexicographic order, and define X(n,m) = Y_n.
then for any (n,m), the set of elements less than (n,m) has a final object: (n,m-1), and so the map colim X(p,q) -> X(n,m) is identified with the isomorphism X(n,m-1) -> X(n,m); that's a cofibration.
however, the colimit over the whole diagram is colim(Y_n). this can't be the hocolim (this construction doesn't even preserve levelwise homotopy equivalences)
if it's, say, a well-order, then you're probably OK because you can show the diagram is cofibrant in the projective model structure
the argument I'm thinking of for that has to induct on "extending lifting, one object of P at a time" and it needs a method for choosing a "next" object of P to do. so maybe you want the poset version of a well-order -- every subset of P has a minimal element?