$\Delta_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta_+)_{{\rm sym}}$ is the symmetric monoidal category generated from the associative operad, this time considered as a symmetric operad. This category can then be explicitly described as the category whose objects are finite sets and such that the morphisms from $I$ to $J$ are given by maps $f:I \to J$ together with, for each $j \in J$, a choice of a linear order on $f^{-1}(i)$. The symmetric monoidal structure is given by disjoint union.

$\Delta_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta_+)_{{\rm sym}}$ is the symmetric monoidal category generated from the associative operad, this time considered as a symmetric operad. This category can then be explicitly described as the category whose objects are finite sets and such that the morphisms from $I$ to $J$ are given by maps $f:I \to J$ together with, for each $j \in J$, a choice of a linear order on $f^{-1}(j)$. The symmetric monoidal structure is given by disjoint union.