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

Algebraically cofibrant and fibrant objects revisited