Proof.
Consider the sequence of morphisms in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) \[\mathrm{Cat}_{\infty}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup} \rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Mod}_{\mathrm{CProj}_k}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}) \simeq \mathrm{add}_k\] left adjoint to the respective forgetful functors (see proposition 3.1.11 for the first two functors, and observation 4.2.4 for the latter one). We will successively lift the (surjective-on-objects, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}\) to \(\mathrm{add}_k\).
Step 1: We first show that the (surjective-on-objects, fully faithful)-functors define a factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup}\) which is of small generation and compatible with the symmetric monoidal structure. Given a morphism \(F\colon \mathcal C\rightarrow\mathcal D\) in \(\mathrm{Cat}_{\infty}^{\sqcup}\), i.e. a functor between \(\infty\)-categories with finite coproducts that preserves finite coproducts, we consider its (surjective-on-objects, fully faithful) factorization in \(\mathrm{Cat}_{\infty}\) 
where \(\widetilde{F}\) is surjective on objects and \(\iota\) is fully faithful. Using observation B.2.1 and lemma B.2.2 we can deduce that the factorization system on \(\mathrm{Cat}_{\infty}\) restricts to one on \(\mathrm{Cat}_{\infty}^{\sqcup}\) which is of small generation and compatible with the symmetric monoidal structure, provided we can show that \(\mathrm{Im}(F)\) admits finite coproducts and that \(\widetilde{F}\) and \(\iota\) preserve them. In fact, since \(\iota\) is fully faithful, it is enough to prove that \(\mathrm{Im}(F)\) is closed under finite coproducts in \(\mathcal D\). To this end, let \(S\) be a finite set and \(I\colon S \rightarrow\mathrm{Im}(F)\) a diagram. Since \(\widetilde{F}\colon \mathcal C\rightarrow\mathrm{Im}(F)\) is surjective on objects, we can lift \(I\) to a functor \(\widetilde{I} \colon S \rightarrow\mathcal C\) which has a colimit \(\mathrm{colim}~\widetilde{I}\in \mathcal C\) by assumption. Since \(F\) preserves coproducts and using the factorization, the coproduct \(\mathrm{colim}~\iota \circ I\) agrees with \(F(\mathrm{colim}~\widetilde{I})\) and hence is in the image of \(F\), as required.
Step 2: To lift from \(\mathrm{Cat}_{\infty}^{\sqcup}\) to \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), observe that the symmetric monoidal left adjoint \((-)^{\mathrm{idem}}:\mathrm{Cat}_{\infty}^{\sqcup} \rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is a reflective localization, i.e. that the right adjoint is fully faithful. We observe the following:
A morphism in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is of the form \((F)^{\mathrm{idem}}\) for a fully faithful morphism \(F\) in \(\mathrm{Cat}_{\infty}^{\sqcup}\) if and only if it is fully faithful.
A morphism in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is of the form \((F)^{\mathrm{idem}}\) for a surjective-on-objects morphism \(F\) in \(\mathrm{Cat}_{\infty}^{\sqcup}\) if and only if it is dominant.
Since the class of dominant functors is stable under retracts in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), it therefore follows from lemma B.2.5 that the (surjective-on-objects, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup}\) induces the (dominant, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), and that this factorization system is of small generation and compatible with the monoidal structure on \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\).
Step 3: Since 
is a monadic adjunction, whose underlying monad \(\mathrm{CProj}_k \otimes - \colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) preserves colimits (and in particular geometric realizations), and preserves dominant functors (since \(\otimes\) is compatible with the (dominant, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), as follows from Step 2), it follows fromlemma B.2.7 that the (dominant, fully faithful) functors form a factorization system on \(\mathrm{add}_k\) which is of small generation and compatible with its symmetric monoidal structure.
Step 4: Lastly, by lemma B.2.3, the (dominant, fully faithful)-factorization system induces one on \(\mathrm{Fun}(B\mathbb{Z}, \mathrm{add}_k)\) which is of small generation and compatible with the Day convolution symmetric monoidal structure. ◻