Dipartimento di Matematica, Universit\`a di Genova, Via Dodecaneso 35, 16146-Genova, Italy
Department of Mathematics and Statistics, York University, Toronto ON, M3J 1P3, Canada
E-mail: firstname.lastname@example.org, email@example.com
Abstract. In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category $\K$ is introduced, as a pair (comonad, monad) over $\K^\bt$. The link with existing notions in terms of morphism classes is given via the respective Eilenberg--Moore categories.