  We extend all known results about transferred model structures on
algebraically cofibrant and fibrant objects by working with weak model
categories. We show that for an accessible weak model category there are always
Quillen equivalent transferred weak model structures on both the categories of
algebraically cofibrant and algebraically fibrant objects. Under additional
assumptions, these transferred weak model structures are shown to be left,
right or Quillen model structures. By combining both constructions, we show
that each combinatorial weak model category is connected, via a chain of
Quillen equivalences, to a combinatorial Quillen model category in which all
objects are fibrant.
