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We show that in a locally -presentable category, every -injectivity class (i.e., the class of all the objects injective with respect to some class of -presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of -presentable morphisms. This was known for small-injectivity classes, and referred to as the ‘small object argument.’ An analogous result is obtained for orthogonality classes and factorization systems, where -filtered colimits play the role of the transfinite compositions in the injectivity case. -presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp. pure-injectives). Finally, locally -presentable categories are shown to be cellularly generated by the set of morphisms between -presentable objects.

$\lambda$ -presentable Morphisms, Injectivity and (Weak) Factorization Systems