  We prove that any right Quillen functor between arbitrary model categories
admits non trivial functorial factorizations that are similar to those of a
model structure. We also prove that these factorizations can be made for lax
monoidal right Quillen functors. Given a monad, operad or a PROP(erad)
$\mathcal{O}$, if we apply one of the factorizations to the forgetful functor
$\mathcal{U} : \mathcal{O}-Alg(\mathcal{M}) \rightarrow \mathcal{M}$, we extend
the theory of Quillen-Segal $\mathcal{O}$-algebras without the hypothesis of
$\mathcal{M}$ being a combinatorial model category.
