Kerodon

Example 9.2.4.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a left adjoint. Then $F$ carries subterminal objects of $\operatorname{\mathcal{C}}$ to subterminal objects of $\operatorname{\mathcal{D}}$, and carries monomorphisms in $\operatorname{\mathcal{C}}$ to monomorphisms in $\operatorname{\mathcal{D}}$. This follows from Proposition 9.2.4.20 and Remark 9.2.2.16, since $F$ preserves limit diagrams (Corollary 7.1.3.21).