Proof of Theorem B.4.1.
Given the factorization system on \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) of Lemma B.4.3, we wish to apply Lemma B.2.5 to the reflective localization 
We note preliminarily that the morphisms in \(\mathrm{Cat}[\mathbb V]\) that are localizations of \(\iota_0\)-equivalences in \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) are precisely the \(\iota_0\)-surjections.
We first show that the hypotheses of Lemma B.2.5 are satisfied. To show that \(RL(\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}) \subseteq \mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}\), we simply observe that if a morphism \(F\) in \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) is homwise in \(\mathcal R\) then so is its localization \(L(F)\) and hence so is \(RL(F)\). To show that \(L(\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}})\) is stable under retracts, it suffices to observe that \(\mathcal L\) is stable under retracts (by definition of a factorization system) and that surjections in \(\mathcal S\) are stable under retracts (since surjections in \(\mathrm{Set}\) are).
From here, the three parts of Lemma B.2.5 respectively imply the three parts of the present result. ◻