In Lurie's `Higher Algebra`

, construction 4.4.2.7 presents a Bar construction in the setting of $\infty$-categories. The construction in 4.4.2.7 takes as input an $\mathcal{O}$-monoidal $\infty$-category $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ and a suitable pair of bi-modules in $\mathcal{C}^\otimes$ and gives a simplicial object in $\mathcal{C}$.

In the proof of the Barr-Beck theorem, this construction is used in lemma 4.7.3.13, where Lurie uses a simplicial object $\mathrm{Bar}_T(T, M)_\bullet$ with values in $\mathrm{LMod}_T(\mathcal{C})$. All of the Bar constructions previously used in Higher-algebra are only defined as taking value in the underlying $\infty$-category.

Looking at the construction, it *feels* obvious that the $\mathcal{C}$-valued simplicial object $\mathrm{Bar}_T(T, M)_\bullet$ should lift to a $\mathrm{LMod}_T(\mathcal{C})$-valued simplicial object, since the bar construction essentially comes from some functor $\mathbf{\Delta}^{op} \to \mathrm{Tens}_{\succ}$ in which the objects should have structure of modules and the maps should be linear, but I am unable to formally write down a lift along the forgetful functor $\mathrm{LMod}_T(\mathcal{C}) \to \mathcal{C}$ from this definition.

Is there a way to see how to to produce such a lift? More generally, given an $(A,B)$-bimodule $M$ in $\mathcal{C}$ and a $(B,C)$-bimodule $N$ in $\mathcal{C}$, the simplicial object $\mathrm{Bar}(M,N)_\bullet$ should take value in the $\infty$-category of $(A,C)$-bimodules, rather than in $\mathcal{C}$. This extension to the bimodule case is also used without proof in HA 5.2.2.6.

This question asks about the definition of $\mathrm{Bar}_T(T, M)_\bullet$ but seems to be only about it as a $\mathcal{C}$-valued object (that's what the answer in the commens gives), and do not address the issue of making it $\mathrm{LMod}_T(\mathcal{C})$-valued.