# Kerodon

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Remark 6.2.3.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ be a functor of $\infty$-categories having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_1 = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$. Suppose that $U$ is a cocartesian fibration, so that the full subcategory $\operatorname{\mathcal{E}}_1 \subseteq \operatorname{\mathcal{E}}$ is reflective (Corollary 6.2.3.2). By virtue of Lemma 6.2.2.10, there exists a $\operatorname{\mathcal{E}}_1$-reflection functor $L: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_1$. Then the restriction $L|_{\operatorname{\mathcal{E}}_0}: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ is given by covariant transport along the unique nondegenerate edge $e$ of $\Delta ^1$ (in the sense of Definition 5.2.2.4). More precisely, if $\eta : \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow L$ is a natural transformation which exhibits $L$ as a $\operatorname{\mathcal{E}}_1$-reflection functor, then $\eta$ carries each object $X \in \operatorname{\mathcal{E}}$ to a $U$-cocartesian morphism $\eta _ X: X \rightarrow L(X)$, so that $\eta$ restricts to a natural transformation $\operatorname{id}_{\operatorname{\mathcal{E}}_0} \rightarrow L|_{\operatorname{\mathcal{E}}_0}$ which witnesses $L|_{\operatorname{\mathcal{E}}_0}$ as given by covariant transport along $e$.

Similarly, if $U$ is a cartesian fibration, then the full subcategory $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ is coreflective; if $L': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_0$ is a $\operatorname{\mathcal{E}}_0$-coreflection functor, then the restriction $L'|_{\operatorname{\mathcal{E}}_1}: \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}_0$ is given by contravariant transport along $e$, in the sense of Definition 5.2.2.14.