Kerodon

Remark 9.4.5.4. Let $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ be a monomorphism of simplicial sets and let $\operatorname{\mathcal{E}}$ be an idempotent complete $\infty $-category. If $\iota $ is a Morita equivalence, then the functor $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ \iota } \operatorname{Fun}(\operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}})$ is both an isofibration (Corollary 4.4.5.3) and an equivalence of $\infty $-categories, and therefore a trivial Kan fibration (Proposition 4.5.5.20). In particular, every diagram $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{E}}$ can be extended to $\operatorname{\mathcal{C}}$.