Kerodon

Warning 4.8.6.14. Remark 4.8.6.13 is generally false in the case $n = -1$, even if we assume that $F$ is an isofibration. For example, let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ be a subcategory. Then the inclusion map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an inner fibration (which is even an isofibration, if $\operatorname{\mathcal{C}}$ is a replete subcategory of $\operatorname{\mathcal{D}}$). The diagonal $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an isomorphism of simplicial sets, and therefore an equivalence of $\infty $-categories. However, $F$ need not be fully faithful.