[Cross-posted from this Math SE question.]
In Higher Algebra, Section 3.3 Lurie constructs the $\infty$-operads $\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\cO}{\mathcal{O}}\newcommand{\cC}{\mathcal{C}}\Mod^{\cO}(\cC)^\otimes$ and $\Mod_A^{\cO}(\cC)^\otimes$ for $\cO^\otimes$ a unital $\infty$-operad, $\cC^\otimes \to \cO^\otimes$ a fibration of (generalized) $\infty$-operads, and $\newcommand{\Alg}{\mathrm{Alg}} A \in \Alg_{/ \cO}(\cC)$ an algebra object. The construction is rather involved: First he constructs a simplicial set $\widetilde{\Mod}^{\cO}(\cC)^\otimes$ together with a map $q\colon \widetilde{\Mod}^{\cO}(\cC)^\otimes \to \cO^\otimes$ such that for all maps of simplicial sets $X \to \cO^\otimes$ there is a canonical bijection $$\newcommand{\Fun}{\mathrm{Fun}} \Fun_{\cO^\otimes}(X, \widetilde{\Mod}^{\cO}(\cC)^\otimes) \cong \Fun_{\Fun(\{1\}, \cO^\otimes)}(X \times_{\Fun(\{0\}, \cO^\otimes)} \mathcal{K}_{\cO}, \cC^\otimes) $$ where $\mathcal{K}_{\cO} \subseteq \Fun(\Delta^1, \cO^\otimes)$ is the full subcategory of semi-inert morphisms in $\cO^\otimes$. He then picks out the full simplicial subset $\overline{\Mod}^{\cO}(\cC)^\otimes \subseteq \widetilde{\Mod}^{\cO}(\cC)^\otimes$ spanned by all vertices $\bar{v}$ such that $\bar{v}$ determines a functor $$ \{v\} \times_{\cO^\otimes} \mathcal{K}_{\cO} \to \cC^\otimes $$ which carries inert morphisms to inert morphisms (here a morphism in $\mathcal{K}_{\cO}$ are inert if its image under the evaluation maps $e_0, e_1\colon \mathcal{K}_{\cO} \to \cO^\otimes$ at the vertices of $\Delta^1$ is inert).
This is followed by around a page of observations, a similar definition for algebras, and finally the definition we care about: In the setting of the first sentence of this question, he defines
$$
\Mod^{\cO}(\cC)^\otimes := \overline{\Mod}^{\cO}(\cC)^\otimes \times_{{}^\mathrm{p} \Alg_{/ \cO}(\cC)}(\cO^\otimes \times \Alg_{/ \cO}(\cC))
$$
and further
$$
\Mod^{\cO}_A(\cC)^\otimes := \Mod^{\cO}(\cC)^\otimes \times_{\Alg_{/ \cO}(\cC)} \{A\}
$$
(I won't describe all maps and constructions involved for brevity's sake).
This definition is, to put it mildly, a bit convoluted. In the introductory paragraph to the section, however, Lurie states that "the construction of $\Mod_A^{\cO}(\cC)^\otimes$ as a simplicial set with a map to $N(\mathrm{Fin}_*)$ is fairly straightforward; the bulk of our work will be in proving that $\Mod^\cO_A(\cC)^\otimes$ is actually an $\infty$-operad." I am thus led to wonder what the "straightforward construction" of $\Mod^\cO_A(\cC)^\otimes$ is supposed to be: Am I to "identify" $\Mod^\cO(\cC)^\otimes$ with $\overline{\Mod}^{\cO}(\cC)^\otimes$? After all, the map $\cO^\otimes \times \Alg_{/ \cO}(\cC) \to {}^\mathrm{p} \Alg_{/ \cO}(\cC)$ I am pulling back along in the definition of the former is a categorical equivalence, so I have a categorical equivalence $\Mod^\cO(\cC)^\otimes \to \overline{\Mod}^{\cO}(\cC)^\otimes$. Even so, the construction of $\overline{\Mod}^{\cO}(\cC)^\otimes$ in and of itself is a little complicated in its own right and not exactly pretty.
My question, thus: Assuming that I don't care about proving that $\Mod^\cO(\cC)^\otimes$ is an $\infty$-operad (or even an $\infty$-category), is there a construction of it that is simpler and/or more concise than the one of $\overline{\Mod}^\cO(\cC)^\otimes$ presented in Higher Algebra, possibly under additional simplifying assumptions?
(Context: I am giving a talk about $\infty$-operads in which I am supposed to introduce, among other things, modules. Giving the full definition would eat a chunk of my time that I already don't have, and any amount of simplification would probably aid understanding in my audience.)
N.b. I have looked at various other sources, e.g. Gepner's An Introduction to Higher Categorical Algebra or Hebestreit's Algebraic and Hermitian $K$-Theory Notes but it seems like everyone either avoids modules altogether or only treats the special case of left/right modules over monoids or modules over commutative monoids.
