Unordered multimorphism notation

Convention 12.3.2. Given a color \(x \in \Oo ^{\simeq }\) and an unordered tuple \(\{y_i\}_{i \in I}\), we define the set \[ \Oo (\{y_i\}_{i \in I}; x) \] of multimorphisms from \(\{y_i\}_{i \in I}\) to \(x\) by choosing a bijection \(I \cong \{1, \dots , n\}\) and using it to order the inputs. The permutation isomorphisms canonically identify the sets obtained from different choices, so the notation is independent of the chosen ordering. Under this convention, the composition map in the operad \(\Oo \) has the form \[ \circ \colon \Oo (\{y_i\}_{i \in I}; x) \times \prod _{i\in I} \Oo (\{z^i_j\}_{j \in J_i}; y_i) \to \Oo (\{z^i_j\}_{(i,j) \in \bigsqcup _{i \in I} J_i};x). \] The permutation isomorphism then says that for every bijection \(\sigma \colon J \xrightarrow {\cong } I\), there is an isomorphism \[ \Oo (\{y_i\}_{i \in I}; x) \xrightarrow {\cong } \Oo (\{y_{\sigma (j)}\}_{j \in J}; x). \]