Lemma 12.4.2. The construction \(\Oo \mapsto \Oo ^{\otimes }\) defines a functor from colored operads to \((2,1)\)-categories over \(\Span (\Fin )\). The \((2,1)\)-category \(\Oo ^{\otimes }\) admits finite products, given by unordered concatenation of tuples, and the structure functor \(\Oo ^{\otimes }\to \Span (\Fin )\) preserves them.
Proof. The operad axioms and the canonical associativity isomorphisms for pullbacks make the composition law unital and associative. A morphism into an unordered concatenation of tuples is uniquely a collection of morphisms into its factors, since both the indexing span and its labeling operations decompose according to the corresponding partition of the target. Thus concatenation satisfies the product universal property. The empty tuple is terminal, and the same description for \(\Comm ^{\otimes }=\Span (\Fin )\) shows that the structure functor preserves these products. □
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