Remark 10.3.5.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories with images, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves pullback diagrams. For any morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, we always have an inclusion $F( \operatorname{im}(f) ) \subseteq \operatorname{im}( F(f) )$ in the partially ordered set $\operatorname{Sub}( F(Y) )$.
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