Kerodon

Example 7.3.5.6. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories and let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the image $G(C) \in \operatorname{\mathcal{E}}$ can be lifted to a $U$-initial object of $\operatorname{\mathcal{D}}$. Applying Proposition 7.3.5.5 (in the special case $\operatorname{\mathcal{C}}^{0} = \emptyset $), we deduce that $G$ can be lifted to a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carries each object of $\operatorname{\mathcal{C}}$ to a $U$-initial object of $\operatorname{\mathcal{D}}$ (see Example 7.3.3.6).