Remark 10.3.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves pullbacks. Then $F$ carries monomorphisms in $\operatorname{\mathcal{C}}$ to monomorphisms in $\operatorname{\mathcal{D}}$ (Proposition 9.3.4.21). In particular, for every object $Y \in \operatorname{\mathcal{C}}$, the functor $F$ carries subobjects of $Y$ to subobjects of $F(Y)$, and therefore induces a map of partially ordered sets $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}( F(Y) )$.
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