  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.
