The stable hull of an exact $\mathrm{\infty }$-category

We construct a left adjoint $$\mathcal{H}$^$\text{st}$\colon $\mathbf{E}\mathbf{x}${\infty} \rightarrow $\mathbf{S}\mathbf{t}${\infty}$totheinclusion$$\mathbf{S}\mathbf{t}${\infty} \hookrightarrow $\mathbf{E}\mathbf{x}${\infty}$ofthe$\infty$-categoryofstable$\infty$-categoriesintothe$\infty$-categoryofexact$\infty$-categories, which we call the stable hull. For every exact $\mathrm{\infty }$-category $\mathcal{E}$, the unit functor $\mathcal{E}\to {\mathcal{H}}^{\text{st}}(\mathcal{E})$ is fully faithful and preserves and reflects exact sequences. This provides an $\mathrm{\infty }$-categorical variant of the Gabriel-Quillen embedding for ordinary exact categories. If $\mathcal{E}$ is an ordinary exact category, the stable hull ${\mathcal{H}}^{\text{st}}(\mathcal{E})$ is equivalent to the bounded derived $\mathrm{\infty }$-category of $\mathcal{E}$.