A note on model (co)slice categories (Q270161)

There are various Quillen adjunctions between model (co)slice categories: the two adjunctions induced by the base change and the cobase change on a model category, and four other adjunctions induced by a Quillen adjunction between two model categories. The author gives a proposition to characterize when these adjunctions are Quillen equivalences. As an application, a triangle equivalence between the stable category of a Frobenius category and the homotopy category of a non-pointed model category is given.