Proof.
Since \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) are module categories by lemma 3.3.5, they inherit via proposition 3.1.8.([0032]) presentably symmetric monoidal structures from the presentably symmetric monoidal categories \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) and \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) (see proposition 3.2.10), respectively. Symmetric monoidality of the functors \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\rightarrow{\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\rightarrow{\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) follows from symmetric monoidality of \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow\mathrm{Pr}^\mathrm{L}\) and \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^\mathrm{L}\). ◻