Lemma 17.3.15. The monoidal unit of \(\Env (\Oo )\) is the empty tuple, and its tensor product is given by concatenation of unordered tuples: \[ \{x_i\}_{i \in I} \otimes \{y_j\}_{j \in J} \simeq \{x_i\}_{i \in I} \sqcup \{y_j\}_{j \in J}. \]
Proof. Under the Segal equivalence, the two tuples on the left determine the object of the fiber over \(\lra {2}\) indexed by their disjoint union \(I \sqcup J\). The tensor product is cocartesian transport of this object along the forward fold span \(\lra {2} \xleftarrow {=} \lra {2} \to \lra {1}\). By the forward-transport calculation in the proof of Proposition 17.3.12, this transport leaves the object of \(\Oo ^\otimes \) unchanged and postcomposes its indexing map with the fold map. The result is the concatenated tuple. The same calculation for the unique forward map \(\emptyset \to \lra {1}\) gives the empty tuple as the unit. □
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