Remark 10.3.3.18 (Functoriality of Images). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{Fun}'( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ be the full subcategory described in Proposition 10.3.3.14. Then $\operatorname{\mathcal{C}}$ has images if and only if the restriction functor
is a trivial Kan fibration. If this condition is satisfied, then $D$ admits a section which carries each morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ to a $2$-simplex
which exhibits $\operatorname{im}(f)$ as an image of $f$. In particular, we can promote the construction $f \mapsto \operatorname{im}(f)$ as a functor of $\infty $-categories $\operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.