ScalingStacks

Remark 2.2. Let $\mathcal{E}\xrightarrow{f}\mathcal{B}$ be a functor of $\infty$-categories. The crucial consequence of a morphism $e_{1}\xrightarrow{\varphi}e_{2}$ in $\mathcal{E}$ being $f$-cocartesian is that for any object $e\in\mathcal{E}$, the induced diagram

is a pullback square in $\mathcal{S}$. Dually, the crucial consequence of a morphism $e_{1}\xrightarrow{\varphi}e_{2}$ in $\mathcal{E}$ being $f$-cartesian is that for any object $e\in\mathcal{E}$, the induced diagram

is a pullback square in $\mathcal{S}$. (Recall Example 1.6, and see [Lur09, Proposition 2.4.1.10].)