Proof.
We begin by fixing a small set \(S\) of morphisms in \(\mathcal C\) that generates \(\mathcal L\). By Proposition B.1.14, we obtain a factorization system \((\mathcal L',\mathcal R')\) on \(\mathcal D\) generated by its image \(F(S)\). So, it remains to show that \(\mathcal L' = U^{-1}(L)\) and that \(\mathcal R' = U^{-1}(R)\).
We first show that \(\mathcal R' = U^{-1}(R)\). For this, note that by definition \(\mathcal R' = F(S)^\perp\). Hence, it suffices to show that a morphism lies in \(F(S)^\perp\) precisely if its image under \(U\) lies in \(\mathcal R\), which follows from the adjunction \(F \dashv U\) (and the fact that \(\mathcal R= S^\perp\)).
We now show that \(\mathcal L' = U^{-1}(\mathcal L)\). For this, let us write \(\mathcal L'' \coloneqq U^{-1}(\mathcal L)\), so that our goal is to show that \(\mathcal L' = \mathcal L''\). Since \(\mathcal L' = \overline{F(S)}\), it is equivalent to show that \(\overline{F(S)} = \mathcal L''\). In other words, it suffices to verify that \(\mathcal L''\) is the smallest saturated class of morphisms in \(\mathcal D\) that contains \(F(S)\).
We deduce this in steps. First of all, the fact that \(\mathcal L''\) contains \(F(S)\) follows from the assumption that the monad \(T\) preserves \(\mathcal L\) and the fact that \(\mathcal L\) contains \(S\).
We now show that \(\mathcal L''\) is saturated by verifying the conditions of Definition B.1.13.
Condition ([00JW]) is clear: \(\mathcal L''\) defines a wide subcategory of \(\mathcal D\).
We now verify condition ([00JX]), i.e. we show that the full subcategory \(\mathcal L'' \subseteq \mathrm{Fun}([1],\mathcal D)\) is closed under small colimits. For this, fix a small \(\infty\)-category \(\mathcal I\) as well as a functor \(\mathcal I\xrightarrow{X} \mathrm{Fun}([1],\mathcal D)\) that factors through \(\mathcal L''\). We wish to show that the colimit \(\mathrm{colim}_\mathcal I(X)\) (computed in \(\mathrm{Fun}([1],\mathcal D)\)) also lies in \(\mathcal L''\), i.e. that \(U(\mathrm{colim}_\mathcal I(X)) \in \mathrm{Fun}([1],\mathcal C)\) lies in \(\mathcal L\). Now, since the adjunction \(F \dashv U\) is monadic, every object \(D \in \mathcal D\) admits a functorial bar resolution: it is the geometric realization of the levelwise free simplicial object \(FT^\bullet U(D) \in \mathrm{Fun}(\Delta^{\mathrm{op}}, \mathcal D)\) [Lur17, Prop. 4.7.3.14]. Using this, we may compute \(\mathrm{colim}_\mathcal I(X)\) as the colimit of the functor 
Namely, we obtain the string of equivalences \[\begin{aligned}
U(\mathrm{colim}_\mathcal I(X))
& \simeq
U(\mathrm{colim}_{\mathcal I\times \Delta^{\mathrm{op}}}(X'))
\\
& \simeq
U( \mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} ( \mathrm{colim}_{i \in \mathcal I} ( X'(i,[n]))))
\\
& \simeq
\mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} U(\mathrm{colim}_{i \in \mathcal I} ( X'(i,[n])))
\\
& \eqqcolon
\mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} U(\mathrm{colim}_{i \in \mathcal I}(FT^nU(X(i))))
\\
& \simeq
\mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} UF(\mathrm{colim}_{i \in \mathcal I}(T^nU(X(i))))
\\
& \eqqcolon
\mathrm{colim}_{[n] \in \Delta^{\mathrm{op}}} T(\mathrm{colim}_{i \in \mathcal I}(T^n U(X(i))))
~,
\end{aligned}\] in which the third equivalence follows from the fact that \(U\) commutes with geometric realizations since \(T\) does by [Lur17, Cor. 4.2.3.5]. Now, by definition of \(\mathcal L'' \coloneqq U^{-1}(\mathcal L)\), for each object \(i \in \mathcal I\) the object \(U(X(i)) \in \mathrm{Fun}([1],\mathcal C)\) lies in \(\mathcal L\). Using repeatedly both the fact that \(T\) preserves \(\mathcal L\) and that \(\mathcal L\subseteq \mathrm{Fun}([1],\mathcal C)\) is closed under colimits, we find that \(U(\mathrm{colim}_\mathcal I(X)) \in \mathrm{Fun}([1],\mathcal C)\) lies in \(\mathcal L\), as desired. So indeed, \(\mathcal L'' \subseteq \mathrm{Fun}([1],\mathcal D)\) is closed under colimits.
The verification of condition ([00JY]) (that \(\mathcal L'' \subseteq \mathrm{Fun}([1],\mathcal D)\) is stable under cobase change) follows from an essentially identical argument (inasmuch as it involves the computation of a colimit (specifically a pushout) in \(\mathrm{Fun}([1],\mathcal D)\)). So indeed, the class \(\mathcal L''\) of morphisms in \(\mathcal D\) is saturated.
In order to conclude that \(\mathcal L'' = \overline{F(S)}\), it therefore remains to show that any saturated class \(\mathcal L'''\) of morphisms in \(\mathcal D\) that contains \(F(S)\) also contains \(\mathcal L''\). Since \(F\) preserves colimits, certainly \(F(\mathcal L) \subseteq \mathcal L'''\). From here, to show the containment \(\mathcal L'' \subseteq \mathcal L'''\), choose any \(f \in \mathcal L'' \coloneqq U^{-1}(\mathcal L)\). Recall that the aforementioned bar resolution yields an equivalence \(|FT^\bullet U(f)| \simeq f\). Note that \(U(f) \in \mathcal L\), and since \(T\) preserves \(\mathcal L\) then \(T^nU(f) \in \mathcal L\), and so all values of the simplicial object \(FT^\bullet U(f)\) lie in \({\sf{F}}(\mathcal L) \subseteq \mathrm{Fun}([1],\mathcal D)\). Hence, its geometric realization – namely, \(f\) – must lie in \(\mathcal L'''\). So indeed, \(\mathcal L''\) is the smallest saturated class of morphisms in \(\mathcal D\) containing \(F(S)\).
It remains to prove the compatibility with \(\mathcal O\)-monoidal structure. Given an operation \((a_1, ..., a_m) \rightarrow b\) in \(\mathcal O\), we want to show that the induced functor \(\mu \colon \mathcal D_{a_1} \times \cdots \times \mathcal D_{a_m} \rightarrow\mathcal D_b\) carries \(\mathcal L'_{a_1} \times \cdots \times \mathcal L'_{a_m}\) to \(\mathcal L'_b\). Explicitly, given morphisms \(X_i \colon [1] \rightarrow\mathcal D_{a_i}\) in \(\mathcal L'_{a_i}\), we would like to show that \(U(\mu(X_1, \cdots, X_m)) \in \mathcal L\). As above, the bar resolution gives us a simplicial object \(X'_i \colon \Delta^{\mathrm{op}}\rightarrow\mathrm{Fun}([1], \mathcal D_{a_i})\) for each \(1 \leq i \leq n\), with \(X'_i([n]) = FT^nU(X_i)\).
We have a string of equivalences: \[ \begin{aligned} U(\mu(X_1, \cdots, X_n)) & \simeq U(\mu( \mathrm{colim}_{[n_1] \in \Delta^{\mathrm{op}}} X'_1([n_1]), \cdots, \mathrm{colim}_{[n_m] \in \Delta^{\mathrm{op}}} X'_m([n_m]))) \\ & \simeq U(\mathrm{colim}_{([n_1], \cdots, [n_m]) \in {(\Delta^{\mathrm{op}})}^m} \mu(X'_1([n_1]), \cdots, X'_m([n_m])))\\ & \simeq U(\mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} \mu(X'_1([n]), \cdots, X'_m([n])))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} U(\mu(X'_1([n]), \cdots, X'_m([n])))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} U(\mu(FT^nU(X_1), \cdots, FT^nU(X_m)))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} UF(\mu(T^nU(X_1), \cdots, T^nU(X_m)))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} T(\mu(T^nU(X_1), \cdots, T^nU(X_m))) ~, \end{aligned}\] in which the third line uses the fact that the diagonal \(\Delta^{\mathrm{op}}\) in \({(\Delta^{\mathrm{op}})}^n\) is cofinal, which is equivalent to the statement that \(\Delta^{\mathrm{op}}\) is sifted [Lur17, Def. 5.5.8.1, Lem. 5.5.8.4]. For \(1 \leq i \leq n\), \(U(X_i) \in \mathcal L\) by asssumption. It follows that \(T^nU(X_i)\) is also in \(\mathcal L\). Since the \(\mathcal O\)-monoidal structure on \(\mathcal C\) is compatible with the factorization system and \(T\) preserves \(\mathcal L\), we see that \(T(\mu(T^nU(X_1), \cdots, T^nU(X_m)))\) is in \(\mathcal L\). The result now follows from ([00L1]) and the fact that \(\mathcal L\) is closed under colimits. ◻