Lemma A.6. ([Anel et al. 2022, Lemma 3.1.8])
Let \((L,R)\) be a factorization system on a category \(C\). Then for every object \(X \in C\) the slice \(C_{/X}\) admits a factorization system in which a morphism lies in the left, resp. right, class if and only if its underlying morphism in \(C\) lies in \(L\), resp. \(R\).
Proof
Given a morphism \(A\to B\) in \(C_{/X}\), factor its underlying morphism in \(C\) as \(A\xrightarrow{l}M\xrightarrow{r}B\). Giving \(M\) the structure map \(M\xrightarrow{r}B\to X\) makes this a factorization in \(C_{/X}\). Orthogonality in the slice follows from orthogonality in \(C\): the unique filler of the underlying square automatically commutes with the structure maps to \(X\).
References
- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. I: Higher sheaves. Adv. Math., 400, 64. 2022.