According to Elmendorf in his paper "Left Adjoints for Generalized Multicategories", generalized multicategories associated to a $\Sigma$-free operad $\mathcal{D}$, which we call $\mathbb{D}$-multicategories after the associated monad $\mathbb{D}$ consists of a set of objects $M_0$, a set of morphisms $M_1$, a target map $T:M_1\to M_0$, a source map $S:M_1\to \mathbb{D}_0M_0$, a unit map $I:M_0\to M_1$, and a composition map which must satisfy some condititons. I will now list these conditions. The unit map must be a presheaf map over the monadic unit map $*\to\mathbb{D}(*)$. The source map must be a map of presheaves over $\mathbb{D}(*)$. The presheaf structure is given by a corollary of a theorem in the paper. The corollary states:
If $X$ supports the structure of presheaf over $\mathbb{D}^k(*)$, then $\mathbb{D}_0X$ supports the structure of presheaf over $\mathbb{D}^{k+1}(*)$.
The target map must be a presheaf map over the terminal functor $\varepsilon:\mathbb{D}(*)\to*$. The morphisms must come equipped with a specified presheaf structure over $\mathbb{D}(*)$. The composition map we require as part of the data of the multicategory $M$ is a map $ \gamma:M_2\to M_1 $ that is a map of presheaves over the monad multiplication $\mu:\mathbb{D}^2(*)\to\mathbb{D}(*)$ where $M_2$ is defined to be the pullback in the following diagram:
$\{M_2\xrightarrow{T} M_1\xrightarrow{S}\mathbb{D}_0M_0\}=\{M_2\xrightarrow{S}\mathbb{D_0}M_1\xrightarrow{\mathbb{D_0}(T)}\mathbb{D_0}M_0\}.$
We note that $\mathbb{D}_0(S):=\coprod_{k\geq 0}(N\mathcal{D}_0)_k\times_{\Sigma_k} S^k$ where $N$ is the nerve functor and where $(N\mathcal{D}_0)_k\times_{\Sigma_k} S^k$ quotients out $(N\mathcal{D}_0)_k\times S^k$ by the relation $(f\cdot\sigma,s)\sim(f,\sigma\cdot s)$ for all $f\in (N\mathcal{D}_0)_k$, $\sigma\in\Sigma_k$, and $ s\in S^k$. We, also, note that $\mathbb{D}\mathcal{C}:=\coprod_{n\ge0}\mathcal{D}_n\times_{\Sigma_n}\mathcal{C}^n$ where we have the same quotient as before.
We now define complete parenthesization by induction as follows:
(1)A list of length 1 is completely parenthesized.
(2)A list of length $n>1$ is completely parenthesized by specifying a $k$ with $1\le k<n$ and a complete parenthesization of the sublists $(s_1,\dots,s_k)$ and $(s_{k+1},\dots,s_n)$.
We need the following lemma:
Let $(s_1,\dots,s_n)$ be a completely parenthesized list, let $1\le j\le n$, and let $(t_1,\dots,t_s)$ be another completely parenthesized list. Then replacing $s_j$ with the list of $t$'s results in a completely parenthesized list of length $n+s-1$.
We now give a definition.
Definition: (The Complete Parenthesization Operad) The set $V_n$ consists of all completely parenthesized lists of the elements of $\{1,\dots,n\}$ in any order. To define the composition, let $(s_1,\dots,s_n):=s$ and $(t_{j1},\dots,t_{jq_j}):=t_j$ be lists for $1\le j\le n$. We need to define $\gamma(s;t_1,\dots,t_n)$. Notice that each $s_j$ is an integer with $1\le s_j\le n$. Then we replace $s_j$ with the list obtained from $t_{s_j}$ by adding $q_1+q_2+\cdots+q_{s_j-1}$ to all entries. For the $\Sigma_n$ action on $V_n$, we just permute the entries in the list, leaving the parenthesization the same. This completes the definition of the operad $V$. We define the complete parenthesization operad to be $EV$ where $E$ is the right adjoint to the set-of-objects functor.
How do I proceed to obtain the category of generalized multicategories of this $\Sigma$-free operad?
I say the following:
We must have that $$\mathbb{D}(*)=\coprod_{n\geq 0}EV_n\times_{\Sigma_n}*.$$ The objects of $\mathbb{D}(*)$ are the disjoint union of the vertices of the associahedra. Each object has a $\Sigma_n$ worth of automorphisms, and there are also a $\Sigma_n$ worth of morphisms between any two objects at level $n$. There are no morphisms between objects at different levels. The monad $\mathbb{D}_0$ is the free magma monad. That is $\mathbb{D}_0X$ is the disjoint union over every natural number $n$ of the set of all completely parenthesized $n$-tuples of $X$.