Definition A.2.
A factorization system on a category \(C\) consists of two classes of morphisms \(L\) and \(R\) of \(C\) satisfying the following two conditions:
Existence of factorization: Every morphism \(f\colon A\to B\) in \(C\) admits a factorization
\[A\xrightarrow{l}X\xrightarrow{r}B\]with \(l\in L\) and \(r\in R\).
Orthogonality: Every morphism \(l \in L\) is left orthogonal to every morphism \(r \in R\).