Lifting PIE limits with strict projections

Abstract: We give a unified direct proof of the lifting of PIE limits to the 2-category of algebras and (pseudo) morphisms, which specifies precisely which of the projections of the lifted limit are strict and detect strictness. In the literature, these limits were lifted one by one, so as to keep track of these projections in each case. We work in the more general context of weak algebra morphisms, so as to include lax morphisms as well. PIE limits are also all simultaneously lifted in this case, provided some specified arrows of the diagram are pseudo morphisms. Again, this unifies the previously known lifting of many particular PIE limits, which were also treated separately.