The following hold.
The symmetric monoidal structure of \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) restricts to a presentably symmetric monoidal structure on the subcategory \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\), which induces a presentably symmetric monoidal structure on the \(\infty\)-category \(\mathrm{add}\) via the equivalence \(\mathcal P^{\Sigma}\colon \mathrm{add}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\).
The symmetric monoidal structure of \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) restricts to a presentably symmetric monoidal structure on the subcategory \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\rightarrow{\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\), which induces a presentably symmetric monoidal structure on the \(\infty\)-category \(\mathrm{st}\) via the equivalence \(\operatorname{Ind}\colon \mathrm{st}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\).