Envelope classical | Stable homotopy theory and higher algebra

Definition 12.3.3. Let $\Oo $ be a colored operad. We define a symmetric monoidal category $\Env (\Oo )$, called the envelope of $\Oo $.

The objects of $\Env (\Oo )$ are unordered tuples $\{x_i\}_{i \in I}$ of colors of $\Oo $;
A morphism $\Phi \colon \{x_i\}_{i \in I} \to \{y_j\}_{j \in J}$ in $\Env (\Oo )$ consists of a morphism of finite sets $f\colon I \to J$ together with a multimorphism $\phi _j \in \Oo (\{x_i\}_{i \in I_j};y_j)$ for every $j \in J$. Here we write $I_j := f^{-1}(j)$ for the preimage of $j$ under $f$. We will visualize this data as follows:
$Five circular nodes $x_1$ to $x_5$ on the left, wired into three rectangular nodes: $x_2$ and $x_3$ enter $\phi_1$, $x_5$ enters $\phi_2$, and $x_1$ and $x_4$ enter $\phi_3$. Each $\phi_j$ has a single output wire, ending in the circular nodes $y_1$, $y_2$ and $y_3$ respectively.$
Here the function $f$ is indicated by the lines from left to right, so that $f(1) = f(4) = 3$, $f(2) = f(3) = 1$ and $f(5) = 2$.
The identity of $\{x_i\}_{i \in I}$ is given by taking $f = \id _I \colon I \to I$, and taking $\phi _j = \id _{x_j} \in \Oo (\{x_i\}_{i \in \{j\}};x_j)$:
$Four parallel rows, none of whose wires cross. In the $i$-th row a circular node $x_i$ is joined by a wire to a rectangular node $\id_{x_i}$, and that node by a further wire to a second circular node $x_i$, for $i = 1, 2, 3, 4$.$
The composition of $\Phi = (f,\phi _j)\colon \{x_i\}_{i \in I} \to \{y_j\}_{j \in J}$ and $\Psi = (g, \psi _k)\colon \{y_j\}_{j \in J} \to \{z_k\}_{k \in K}$ is given by $\Psi \circ \Phi = (h, \xi _k)$ where $h := g \circ f\colon I \to K$ is the composition of $g$ and $f$, and for every $k \in K$ the multimorphism $\xi _k \in \Oo (\{x_i\}_{i \in I_k};z_k)$ is the image of $(\psi _k, (\phi _j)_{j \in J_k})$ under the composition map \[ \circ \colon \Oo (\{y_j\}_{j \in J_k};z_k) \times \prod _{j \in J_k} \Oo (\{x_i\}_{i \in I_j}; y_j) \to \Oo (\{x_i\}_{i \in I_k};z_k); \] here we are implicitly using that $I_k = (g \circ f)^{-1}(k) = f^{-1}(g^{-1}(k))$ may be written as the disjoint union of the sets $I_j = f^{-1}(j)$ for $j \in J_k = g^{-1}(k)$.
The fact that composition in $\Env (\Oo )$ is unital and associative precisely amounts to the unitality and associativity conditions for colored operads.

The category $\Env (\Oo )$ admits a symmetric monoidal structure given by unordered concatenation of tuples.

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