Warning 10.3.2.10. Let $f: X \rightarrow Y$ be a quotient morphism in an $\infty $-category $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ is not (the nerve of) an ordinary category, then $f$ need not be an epimorphism. For example, let $(X,x)$ be a pointed Kan complex, and let $\iota : \{ x\} \rightarrow X$ denote the inclusion map, which we regard as a morphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces. Then:
The morphism $\iota $ is an epimorphism (in the $\infty $-category $\operatorname{\mathcal{S}}$) if and only if $X$ is contractible. To see this, we observe that the identity map ${\operatorname{id}}_{X}$ and the constant map $c: X \rightarrow \{ x\} \xrightarrow {\iota } X$ become homotopic after precomposition with $\iota $; if $\iota $ is an epimorphism, it follows that ${\operatorname{id}}_{X}$ is homotopic to $c$.
The morphism $\iota $ is a quotient morphism if and only if $X$ is connected (see Proposition 10.3.4.17).