Quillen adjunction

Remarks

For an adjunction, $F(i)$ has the left lifting property against $p$ if and only if $i$ has the left lifting property against $G(p)$. Since the classes of a model structure are characterized by lifting properties, the three conditions are equivalent.

A Quillen adjunction induces an adjunction $\mathbf{L}F \dashv \mathbf{R}G$ between the homotopy categories. Here $\mathbf{L}F$ is computed by applying $F$ to a cofibrant replacement, and $\mathbf{R}G$ by applying $G$ to a fibrant replacement. This works because $F$ sends weak equivalences between cofibrant objects to weak equivalences, by Ken Brown's lemma.

Example: geometric realization $\abs{-}$ from simplicial sets to topological spaces is left adjoint to the singular simplicial set functor $\mathrm{Sing}$. With the Kan–Quillen and Quillen model structures, $\mathrm{Sing}$ sends Serre fibrations to Kan fibrations and trivial ones to trivial ones, so the pair is a Quillen adjunction.

Non-example: let $\mathcal{M}$ and $\mathcal{N}$ both be the category of sets. Give $\mathcal{M}$ the trivial model structure: $W$ the bijections, and all maps fibrations and cofibrations. Give $\mathcal{N}$ the model structure with $W$ all maps, cofibrations the bijections, and fibrations all maps. The identity adjunction fails clause 1, since a non-bijective map is a cofibration of $\mathcal{M}$ but not of $\mathcal{N}$.