Closure properties of factorization systems

The inclusions \(L\subseteq{}^{\perp}R\) and \(R\subseteq L^{\perp}\) are part of the definition. Conversely, suppose that \(f\in{}^{\perp}R\) and choose a factorization of \(f\) into \(l\in L\) followed by \(r\in R\). Orthogonality of \(f\) and \(r\) gives a section \(s\) of \(r\). Comparing the endomorphism \(sr\) of the middle object with its identity, using \(l\perp r\), shows that \(s\) is also a retraction. Thus \(r\) is an isomorphism and \(f\in L\). This proves \(L={}^\perp R\); the proof that \(R=L^\perp\) is dual.Orthogonality classes contain the isomorphisms and are closed under composition and retracts, which proves (2) and (3). A lifting problem against a base change of a morphism in \(R\) is equivalently a lifting problem against the original morphism; hence \(R\) is closed under base change. Dually, \(L\) is closed under cobase change. The cancellation properties follow by applying orthogonality successively to the two composable morphisms.Finally, if \((r_i)_i\) is a small diagram of morphisms in \(R\), then for every \(l\in L\) the anima of fillers from \(l\) to \(\lim_i r_i\) is the limit of the contractible animae of fillers from \(l\) to the \(r_i\). It is therefore contractible, so \(\lim_i r_i\in R\). The colimit statement for \(L\) is dual. Compare [Lurie 2009, Propositions 5.2.8.6 and 5.2.8.11].