Proof.
We show that the composite \((-)^{\mathrm{fin}}\) is indeed left adjoint to the forgetful functor. By construction, we have a commutative diagram in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) 
where the top horizontal morphism is the functor from construction 3.4.4, and the bottom horizontal functor is the subcategory inclusion (which is a symmetric monoidal left adjoint by proposition 3.2.10). Taking right adjoints, the middle square of functors in the following diagram commutes: 
The left and right square commute by proposition 3.3.4. By proposition 3.2.8.([003Z]), the bottom horizontal composite is the forgetful functor \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\). The top horizontal functor is the right adjoint to \((-)^{\mathrm{fin}}\) and hence agrees with the forgetful functor \(\mathrm{st}\rightarrow\mathrm{add}\).
We next prove fully faithfulness of the unit: Since both are left adjoints of the forgetful functor, the functor ([005B]) is equivalent to the functor \((-)^{\mathrm{fin}}\colon \mathrm{add}\rightarrow\mathrm{st}\) constructed in [ES22, Def. 2.1.17] which sends an additive, idempotent-complete \(\infty\)-category \(\mathcal C\) to the smallest full stable subcategory of \(\mathrm{Fun}^{\times}(\mathcal C^\mathrm{op}, \mathrm{Sp})\) (the category of functors taking finite coproducts in \(\mathcal C\) to finite products in \(\mathrm{Sp}\)) containing the image of the Yoneda embedding. In [ES22, Cor. 2.1.5], it is shown that the inclusion \(\mathcal C\rightarrow\mathrm{Fun}^{\times}(\mathcal C^{\mathrm{op}}, \mathrm{Sp})\) is fully faithful, and hence so is the inclusion \(\mathcal C\rightarrow\mathcal C^{\mathrm{fin}}\). ◻