ScalingStacks

Proof. Let ${\mathcal{C}}$ be a presentable stable $\infty$-category, and note that a colimit-preserving functor ${\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ sends the arrows in $S$ to equivalences in ${\mathcal{C}}$ if and only if its restriction to ${\mathcal{A}}$ is trivial. We therefore may identify

$$\mathrm{Fun}^{\mathrm{L}}({\mathcal{B}}/{\mathcal{A}},{\mathcal{C}})\subseteq\mathrm{Fun}^{\mathrm{L}}({\mathcal{B}},{\mathcal{C}}),$$

with the full subcategory spanned by those colimit-preserving functors ${\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ which send the arrows in $S$ to equivalences in ${\mathcal{C}}$. It follows from lemma 5.5 that ${\mathcal{B}}/{\mathcal{A}}\simeq T^{-1}{\mathcal{B}}$, where $T$ is the strongly saturated class of arrows of ${\mathcal{B}}$ which become equivalences in ${\mathcal{B}}/{\mathcal{A}}$.

We now show that $S$ is strongly saturated, so that $S=T$. First, suppose given a cofiber sequence $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z$ in ${\mathcal{B}}$ such that $Z$ lies in the essential image of ${\mathcal{A}}$, and let $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X^{\prime}$ be any map. Then the cofiber of $X^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X^{\prime}\coprod_{X}Y$ is equivalence to $Z$, so is also in the essential image of ${\mathcal{A}}$. Second, given a diagram $f_{\alpha}\colon X_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y_{\alpha}$ in $\mathrm{Fun}(\Delta^{1},{\mathcal{B}})$ with colimit $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$, and suppose that the cofibers $Z_{\alpha}$ of each $f_{\alpha}$ lies in the essential image of ${\mathcal{A}}$. Commuting colimits implies that the cofiber $Z$ of $f$ is computed as the colimit of the $Z_{\alpha}$, and this lies in the essential image of ${\mathcal{A}}$ since ${\mathcal{A}}$ is closed under colimits and the functor ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ preserves colimits. Lastly, suppose $h=g\circ f$ is a composite of $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$ followed by $g\colon Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z$, and write $Y/X$, $Z/Y$, and $Z/X$ for the cofibers of $f$, $g$, and $h$, respectively. Then we have a cofiber sequence $Y/X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z/X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z/Y$, so if any two lie in the essential image of ${\mathcal{A}}$ then so does the third. ∎