Proposition 10.3.3.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ has images, the following conditions are equivalent:
- $(1)$
The morphism $f$ is a quotient morphism.
- $(2)$
The morphism $f$ is left orthogonal to every monomorphism in $\operatorname{\mathcal{C}}$.
- $(3)$
The morphism $f$ is weakly left orthogonal to every monomorphism in $\operatorname{\mathcal{C}}$.