Finite-stage equality in a filtered set colimit ($[i,x]=[j,y]⟺D(u)x=D(v)y\text{ at some common stage}$)

In a filtered colimit in a category of sets, any finite zigzag of generating identifications can be moved to one common stage. The filtered category conditions supply common objects and equalize the finitely many competing arrows. Consequently finitely many representatives and finitely many equalities can be realized simultaneously at one stage. This is the elementwise reason filtered colimits commute with finite limits in sets.