ScalingStacks

Remark 2.3. A cocartesian fibration $\mathcal{E}\xrightarrow{f}\mathcal{B}$ whose fibers $f^{-1}(b)$ are all $\infty$-groupoids is called a left fibration. These assemble into the full subcategory $\mathcal{LF}\textup{ib}(\mathcal{B})\subset\textup{co}\mathcal{C}\mathcal{F}\textup{ib}(\mathcal{B})$. Correspondingly, the (covariant) Grothendieck construction restricts to an equivalence

from the $\infty$-category of functors $\mathcal{B}\rightarrow\mathcal{S}$ valued in the $\infty$-category of spaces to the $\infty$-category $\mathcal{LF}\textup{ib}(\mathcal{B})$ of left fibrations over $\mathcal{B}$. As a result of the fact that all morphisms in a space are equivalences, given a left fibration $\mathcal{E}\xrightarrow{f}\mathcal{B}$, every morphism in $\mathcal{E}$ is $f$-cocartesian.

Dually, a cartesian fibration whose fibers are all $\infty$-groupoids is called a right fibration, the contravariant Grothendieck construction restricts to an equivalence

from the $\infty$-category of functors $\mathcal{B}^{op}\rightarrow\mathcal{S}$ to the full subcategory $\mathcal{RF}\textup{ib}(\mathcal{B})\subset\mathcal{C}\mathcal{F}\textup{ib}(\mathcal{B})$ of right fibrations over $\mathcal{B}$, and given a right fibration $\mathcal{E}\xrightarrow{f}\mathcal{B}$, every morphism in $\mathcal{E}$ is $f$-cartesian.