Example 4.3.3.3. Let $E: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote the functor given by
\[ E(I) = \begin{cases} \ast & \textnormal{ if $I = \emptyset $} \\ \emptyset & \textnormal{ otherwise. } \end{cases} \]
For every functor $X: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$, we have canonical bijections
\[ (X \star E)(J) = \coprod _{I \sqsubseteq J} (X(I) \times E( J \setminus I)) \simeq X(J) \times E(\emptyset ) \simeq X(J) \]
\[ (E \star X)(J) = \coprod _{I \sqsubseteq J} (E(I) \times X(J \setminus I)) \simeq E(\emptyset ) \times X(J) \simeq X(J). \]
These bijections depend functorially on $J$, and therefore determine isomorphisms of functors
\[ X \star E \simeq X \simeq E \star X. \]