Remark 6.3.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a localization functor between (small) $\infty $-categories. Then $F$ is an epimorphism in the $\infty $-category $\operatorname{\mathcal{QC}}$ (see Variant 4.7.5.3). Beware that the converse is false in general. For example, suppose that $\operatorname{\mathcal{C}}$ is a Kan complex. Then:
The projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is a localization functor if and only if $\operatorname{\mathcal{C}}$ is contractible.
The projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is an epimorphism (in the $\infty $-category $\operatorname{\mathcal{QC}}$) if and only if $\operatorname{\mathcal{C}}$ is acyclic: that is, it is connected and the homology groups $\mathrm{H}_{\ast }(X, \operatorname{\mathbf{Z}})$ vanish for $\ast > 0$.