A functor \(F \colon \mathcal{C} \to \mathcal{D}\) is a left Quillen functor if it is a left adjoint that preserves cofibrations and trivial cofibrations.
A functor \(U \colon \mathcal{D} \to \mathcal{C}\) is a right Quillen functor if it is a right adjoint that preserves fibrations and trivial fibrations.
An adjunction \(F \colon \mathcal{C} \rightleftarrows \mathcal{D} \colon U\) is a Quillen adjunction if \(F\) is left Quillen — equivalently, if \(U\) is right Quillen.
A Quillen adjunction \(F \dashv U\) is a Quillen equivalence if for every cofibrant \(X \in \mathcal{C}\) and fibrant \(Y \in \mathcal{D}\text{,}\) a map \(FX \to Y\) is a weak equivalence in \(\mathcal{D}\) if and only if its adjoint \(X \to UY\) is a weak equivalence in \(\mathcal{C}\text{.}\)
