Remark 4.3.3.5 (Associativity). Let $X$, $Y$, and $Z$ be functors from $\operatorname{Lin}^{\operatorname{op}}$ to the category of sets. For every finite linearly ordered set $K$, we have a canonical bijection
\begin{eqnarray*} (X \star (Y \star Z))(K) & = & \coprod _{ I \sqsubseteq K} (X(I) \times (Y \star Z)(K \setminus I) ) \\ & = & \coprod _{I \sqsubseteq K} (X(I) \times \coprod _{J \sqsubseteq K \setminus I} (Y(J) \times Z(K \setminus (I \cup J)) ) ) \\ & \simeq & \coprod _{I \sqsubseteq K} \coprod _{J \sqsubseteq K \setminus I} (X(I) \times Y(J) \times Z(K \setminus (I \cup J) ) ) \\ & \simeq & \coprod _{J' \sqsubseteq K} \coprod _{I \sqsubseteq J'} (X(I) \times Y(J' \setminus I) \times Z(K \setminus J') ) \\ & \simeq & \coprod _{J' \sqsubseteq K} ( \coprod _{I \sqsubseteq J'} (X(I) \times Y(J' \setminus I) \times Z(K \setminus J' ) ) \\ & = & \coprod _{J \sqsubseteq K} ((X \star Y)(J') \star Z(K \setminus J' )) \\ & = & ((X \star Y) \star Z)(K). \end{eqnarray*}
These bijections depend functorially on $K \in \operatorname{Lin}^{\operatorname{op}}$, and therefore supply an isomorphism of functors $\alpha _{X,Y,Z}: X \star (Y \star Z) \simeq (X \star Y) \star Z$.