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Proposition 3.2.10.
The symmetric monoidal structure of \(\mathrm{Pr}^\mathrm{L}\) restricts to presentably symmetric monoidal structures on the subcategories \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) and \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) of \(\mathrm{Pr}^\mathrm{L}\).
Together with the symmetric monoidal left adjoint of the forgetful functor \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) from proposition 3.1.11.([003D]), the symmetric monoidal subcategory inclusion \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) assemble into a commutative diagram in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\): 
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Proof.
Recall from proposition 3.1.11 the functor \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) which is left adjoint to the forgetful functor. By proposition 3.2.8.([003Z]), this functor is equivalent to the composite \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\simeq \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\simeq \mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) for the subcategory inclusion \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\). Hence, this induces presentably symmetric monoidal structures on \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) and \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) and on the inclusion \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\).
It remains to show that the subcategory inclusion \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow\mathrm{Pr}^\mathrm{L}\) is symmetric monoidal which follows since the composite \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\xrightarrow{\operatorname{Ind}} \mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow
\mathrm{Pr}^\mathrm{L}\) is [Lur17, Lem. 5.3.2.11]. ◻