$\mathbb{E}_M$ as colimit of little cubes operads In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}_M$ can be identified with the colimit of a diagram of $\infty$-operads parametrized by $M$, each of which is equivalent to $\mathbb{E}_k$". I do not see how this follows from the preceding discussion there, but let me give a bit of context:
Lurie introduces the $\mathbb{E}_M$-operad of a topological manifold $M^n$ by forming the pullback $\operatorname{BTop}(n)^\otimes \times_{\operatorname{BTop}(n)^\coprod} B_M^\coprod$, where $B_M$ is defined as the category $\operatorname{BTop}(n)_{/M}$ with $\operatorname{BTop(n)}$ the full sub-$\infty$-category on $\mathbb{R}^n$ of the category of topological manifolds and embeddings. Note that the operadic structures are the coCartesian symmetric monoidal structure $\coprod$, and disjoint unions, respectively (i.e. $\operatorname{Mul}_{\operatorname{BTop}(n)^\otimes}(\mathbb{R}^n,\mathbb{R}^n ; \mathbb{R}^n) = \operatorname{Emb}(\mathbb{R}^n \sqcup \mathbb{R}^n, \mathbb{R}^n)$ for example). To put it bluntly, $\mathbb{E}_M$ describes embeddings of (disjoint unions of) disks into $M$.
In the previously mentioned remark, Lurie shows that $B_M \simeq \operatorname{Sing}(M)$ so that, since $\operatorname{BTop}(n)^\otimes$ has only one object $\mathbb{R}^n$, the underlying $\infty$-category of $\mathbb{E}_M$ is equivalent to $\operatorname{Sing}(M)$. It is also not very hard to see that if we take the full sub-operad of $\mathbb{E}_M$ on one object, i.e. one point in $M$, it is equivalent to $\mathbb{E}_n$. However, I do not see how the statement about colimits above follows from these observations, as Lurie claims - I however find the statement very interesting, as it is in my eyes much more explicit then the assembly of $\mathbb{E}_M$ by copies of $\mathbb{E}_k$ that Lurie proves later.
 A: I agree this is sort of scattered around as remarks (e.g. HA.2.3.3.4, say). Let's try to spell it out more cleanly.

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*I claim that, if $X$ is a groupoid, then the category of $X$-families of operads is the same as the category of functors $\mathsf{Fun}(X, \mathsf{Op})$. Indeed, $\mathsf{Op}$ is a (non-full) subcategory of $\mathsf{Cat}\downarrow \mathsf{Fin}_*$, so begin by observing that $\mathsf{Fun}(X,\mathsf{Cat}\downarrow \mathsf{Fin}_*)$ is equivalent, by the Grothendieck construction, to the category of diagrams $X \leftarrow \mathcal{E} \to \mathsf{Fin}_*$ where $\mathcal{E} \to X$ is a cocartesian fibration. But if $X$ is a groupoid, every functor with target $X$ is a cocartesian fibration (here I am using the 'homotopy invariant notions', otherwise you have to say that we restrict to maps which are categorical fibrations, etc.) So in fact, $\mathsf{Fun}(X, \mathsf{Cat}\downarrow \mathsf{Fin}_*)\simeq \mathsf{Cat}\downarrow(X \times \mathsf{Fin}_*)$. But now the definition of an $X$-family of operads corresponds exactly to the requirement that the functor from $X$ to $\mathsf{Cat}\downarrow \mathsf{Fin}_*$ factors through the subcategory of operads.

*Great. Let $\mathcal{E}$ be a generalized operad and $\mathcal{O}$ be an operad. By definition, a map $\mathrm{Assem}(\mathcal{E}) \to \mathcal{O}$ is the same as a map of generalized operads $\mathcal{E} \to \mathcal{O}$. This, in turn, is the same as a map of generalized operads over $\mathcal{E}_{\langle 0\rangle}\times \mathsf{Fin}_*$ from $\mathcal{E}$ to $\mathcal{E}_{\langle 0\rangle}\times  \mathcal{O}$. By HA.2.3.2.13, this is the same as a map of families of operads over $\mathcal{E}_{\langle 0\rangle}$ from $\mathcal{E} \to \mathcal{E}_{\langle 0\rangle} \times \mathcal{O}$. Finally, if $\mathcal{E}_{\langle 0\rangle}$ happens to be a groupoid, then by (1) this is the same as a map from the diagram of operads classifying $\mathcal{E}$ to the constant diagram at $\mathcal{O}$. This completes the proof that $\mathrm{Assem}(\mathcal{E})$ is the colimit of the corresponding diagram (when $\mathcal{E}_{\langle 0\rangle}$ happens to be a groupoid.)

