Kerodon

Warning 10.3.3.8 (Essential Images). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Recall that the essential image of $F$ is the full subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ spanned by objects $D \in \operatorname{\mathcal{D}}$ which are isomorphic to $F(C)$, for some object $C \in \operatorname{\mathcal{C}}$ (Definition 4.6.2.12). In this case, the inclusion map $\iota : \operatorname{\mathcal{D}}_0 \hookrightarrow \operatorname{\mathcal{D}}$ is always a monomorphism in $\operatorname{\mathcal{QC}}$ (Corollary 9.3.4.34), and $F$ factors (uniquely) as the composition of $\iota $ with a functor $F_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_0$. Beware that this factorization generally does not exhibit $\operatorname{\mathcal{D}}_0$ as an image of $F$ in the $\infty $-category $\operatorname{\mathcal{QC}}$, in the sense of Definition 10.3.3.1: that is, the functor $F_0$ need not be a quotient morphism in $\operatorname{\mathcal{QC}}$. This fails, for example, if $F$ is the inclusion functor $\operatorname{\partial \Delta }^{1} \hookrightarrow \Delta ^1$.