Factorization System - Orthogonality

Orthogonality

Two morphisms and are said to be orthogonal, denoted, if for every pair of morphisms and such that there is a unique morphism such that the diagram

commutes. This notion can be extended to define the orthogonals of sets of morphisms by

and

Since in a factorization system contains all the isomorphisms, the condition (3) of the definition is equivalent to

(3') and