# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Lemma 6.2.3.1. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category equipped with a functor $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_1 = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$. Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{E}}$. Then:

• The morphism $f$ exhibits $X$ as a $\operatorname{\mathcal{E}}_0$-coreflection of $Y$ (in the sense of Definition 6.2.2.1) if and only if $X$ belongs to $\operatorname{\mathcal{E}}_0$ and $f$ is $\pi$-cartesian.

• The morphism $f$ exhibits $Y$ as a $\operatorname{\mathcal{E}}_1$-reflection of $X$ if and only if $Y$ belongs to $\operatorname{\mathcal{E}}_1$ and $f$ is $\pi$-cocartesian.

Proof. This is a special case of Corollary 5.1.2.3. $\square$