ScalingStacks

Definition 5.18. An exact sequence of small $\kappa$-cocomplete stable $\infty$-categories and $\kappa$-small colimit preserving functors

is called split-exact if there exist exact functors $i\colon{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}$ and $j\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$, right adjoint to $f$ and $g$, respectively, such that $i\circ f\simeq\Id$ and $g\circ j\simeq\Id$ via the adjunction morphisms.