there exists a unique diagonal filler making both triangles commute:
Given a class of maps , the class is denoted or . Likewise, given , the class is denoted or . These operations form a Galois connection on the poset of classes of morphisms in the ambient category. In particular, we have and .
A pair such that and is sometimes called a prefactorization system. If in addition every morphism factors as an -morphism followed by an -morphism, it is an (orthogonal) factorization system.
A strong epimorphism in any category is, by definition, an epimorphism in , where is the class of monomorphisms. (If the category has equalizers, then every map in is epic.) Dually, a strong monomorphism is a monomorphism in .
The orthogonal subcategory problem is related to localization. Suppose is indeed a reflective subcategory; let be the reflector (the left adjoint to the inclusion ). Certainly sends arrows in to isomorphisms in . Indeed, if belongs to , then the inverse to is the unique arrow extending along to an arrow , using the fact that belongs to .