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Edit — some clarifications in answer to OP’s comments: To see why the data above generates a map of $\omega$-categories $\mathcal{O}(n) \to {\boxtimes}^n$, we rely on two facts:

• The universal property of $\mathcal{O}_n$ as “freely generated” by its atomic cells, i.e. the cells of the parity complex: roughly to give a map from $\newcommand{\O}{\mathcal{O}}\O_n$ to any $\omega$-category, it suffices to define the map just on atomic cells, preserving the source and target. This universal property is defined and proved for $\O_n$ in Street 1986, The algebra of oriented simplices, §4 (unpaywalled scan), and generalised to arbitrary parity complexes in Street 1991 Parity complexes, §4.

• The descriptions of the cells of the $\omega$-category $\boxtimes^n$ as suitable finite sets of atomic cells, and the calculations of their sources, targets, and composition. This construction is presented for arbitrary parity complexes in Street 1991 §3, and the specific parity complex for $\boxtimes^n$ is described in Aitchison 1986/2010.

Edit — some clarifications in answer to OP’s comments: To see why the data above generates a map of $\omega$-categories $\mathcal{O}_n \to {\boxtimes}^n$, we rely on two facts:

• The universal property of $\mathcal{O}_n$ as “freely generated” by its atomic cells, i.e. the cells of the parity complex: roughly to give a map from $\newcommand{\O}{\mathcal{O}}\O_n$ to any $\omega$-category, it suffices to define the map just on atomic cells, preserving the source and target. This universal property is defined and proved for $\O_n$ in Street 1986, The algebra of oriented simplices, §4 (unpaywalled scan), and generalised to arbitrary parity complexes in Street 1991 Parity complexes, §4. Simon Forest showed (summarised here, with link to full paper) that Street’s general version of the theorem is wrong, but notes that it holds for most well-known examples, in particular for $\O_n$.

• The descriptions of the cells of the $\omega$-category $\boxtimes^n$ as suitable finite sets of atomic cells, and the calculations of their sources, targets, and composition. This construction is presented for arbitrary parity complexes in Street 1991 §3, and the specific parity complex for $\boxtimes^n$ is described in Aitchison 1986/2010.

• The universal property of $\mathcal{O}_n$ as “freely generated” by its atomic cells, i.e. the cells of the parity complex: roughly to give a map from $\newcommand{\O}{\mathcal{O}}\O_n$ to any $\omega$-category, it suffices to define the map just on atomic cells, preserving the source and target. This universal property is defined and proved for $\O_n$ in Street 1986, The algebra of oriented simplices, §4 (and generalised to arbitrary parity complexes in Street 1991 Parity complexes, §4).

• The descriptions of the cells of the $\omega$-category $\boxtimes^n$ as suitable finite sets of atomic cells, and the calculations of their sources, targets, and composition. This construction is presented for arbitrary parity complexes in Street 1991 §3, and the specific parity complex for $\boxtimes^n$ is described in Aitchison 1986/2010.

• The universal property of $\mathcal{O}_n$ as “freely generated” by its atomic cells, i.e. the cells of the parity complex: roughly to give a map from $\newcommand{\O}{\mathcal{O}}\O_n$ to any $\omega$-category, it suffices to define the map just on atomic cells, preserving the source and target. This universal property is defined and proved for $\O_n$ in Street 1986, The algebra of oriented simplices, §4 (unpaywalled scan), and generalised to arbitrary parity complexes in Street 1991 Parity complexes, §4.

• The descriptions of the cells of the $\omega$-category $\boxtimes^n$ as suitable finite sets of atomic cells, and the calculations of their sources, targets, and composition. This construction is presented for arbitrary parity complexes in Street 1991 §3, and the specific parity complex for $\boxtimes^n$ is described in Aitchison 1986/2010.

Edit — some clarifications in answer to OP’s comments: To see why the data above generates a map of $\omega$-categories $\mathcal{O}(n) \to {\boxtimes}^n$, we rely on two facts:

• The universal property of $\mathcal{O}_n$ as “freely generated” by its atomic cells, i.e. the cells of the parity complex: roughly to give a map from $\newcommand{\O}{\mathcal{O}}\O_n$ to any $\omega$-category, it suffices to define the map just on atomic cells, preserving the source and target. This universal property is defined and proved for $\O_n$ in Street 1986, The algebra of oriented simplices, §4 (and generalised to arbitrary parity complexes in Street 1991 Parity complexes, §4).

• The descriptions of the cells of the $\omega$-category $\boxtimes^n$ as suitable finite sets of atomic cells, and the calculations of their sources, targets, and composition. This construction is presented for arbitrary parity complexes in Street 1991 §3, and the specific parity complex for $\boxtimes^n$ is described in Aitchison 1986/2010.