ScalingStacks

Proof. The fully faithful inclusions ${\mathcal{A}}\subset\Ind_{\kappa}({\mathcal{A}})$ and ${\mathcal{B}}\subset\Ind_{\kappa}({\mathcal{B}})$ show that ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is fully faithful if and only if $\Ind_{\kappa}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\kappa}({\mathcal{B}})$ is fully faithful (for the reverse direction, this follows from the definition of the mapping spaces in $\Ind_{\kappa}(-)$). Thus it remains to check that ${\mathcal{B}}/{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ is an equivalence upon idempotent completion if and only if $\Ind_{\kappa}({\mathcal{B}})/\Ind_{\kappa}({\mathcal{A}})\simeq\Ind_{\kappa}({\mathcal{B}}/{\mathcal{A}})$. Since $\Ind_{\kappa}$ preserves cofibers, it is enough to check that the equivalence $\Ind_{\kappa}({\mathcal{B}})/\Ind_{\kappa}({\mathcal{A}})\simeq\Ind_{\kappa}({\mathcal{B}}/{\mathcal{A}})$ implies the equivalence ${\mathcal{B}}/{\mathcal{A}}\simeq{\mathcal{C}}$ whenever the latter are idempotent complete. Thus, given a $\kappa$-cocomplete small stable $\infty$-category ${\mathcal{D}}$ (which we assume is idempotent complete if $\kappa=\omega$), we must show that

$$\mathrm{Fun}^{\ex(\kappa)}({\mathcal{C}},{\mathcal{D}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex(\kappa)}({\mathcal{B}},{\mathcal{D}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex(\kappa)}({\mathcal{A}},{\mathcal{D}})$$

is a fiber sequence of $\infty$-categories. Since ${\mathcal{D}}\simeq(\Ind_{\kappa}({\mathcal{D}}))^{\kappa}$, by adjunction this is equivalent to the sequence

$$\mathrm{Fun}^{\mathrm{L}}(\Ind_{\kappa}({\mathcal{C}}),\Ind_{\kappa}({\mathcal{D}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\mathrm{L}}(\Ind_{\kappa}({\mathcal{B}}),\Ind_{\kappa}({\mathcal{D}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\mathrm{L}}(\Ind_{\kappa}({\mathcal{A}}),\Ind_{\kappa}({\mathcal{D}})),$$

which is a fiber sequence by assumption. ∎