Exercise 9.2.9.10. Let $\operatorname{Set}$ denote the category of sets and let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{Set})$ be the associated $\infty $-category. Let $S$ be the collection of surjective functions, and let $I$ be the collection of injective functions. Show that:
The pair $(S, I )$ is a factorization system on $\operatorname{\mathcal{C}}$.
The pair $(I,S)$ is a weak factorization system on $\operatorname{\mathcal{C}}$.
The pair $(I,S)$ is not a factorization system on $\operatorname{\mathcal{C}}$.