\[ \text{id}^{\otimes }_{-_{1},-_{2}}\colon -_{1}\otimes _{\mathcal{C}}-_{2}\to -_{1}\boxtimes _{\mathcal{C}}-_{2} \]
be a natural transformation.
-
If there exists a natural transformation
\[ \alpha ^{\otimes }_{A,B,C}\colon \webleft (A\otimes _{\mathcal{C}}B\webright )\boxtimes _{\mathcal{C}}C\to A\otimes _{\mathcal{C}}\webleft (B\boxtimes _{\mathcal{C}}C\webright ) \]
making the diagrams
and
commute, then the natural transformation $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.
-
If there exists a natural transformation
\[ \alpha ^{\boxtimes }_{A,B,C}\colon \webleft (A\boxtimes _{\mathcal{C}}B\webright )\otimes _{\mathcal{C}}C\to A\boxtimes _{\mathcal{C}}\webleft (B\otimes _{\mathcal{C}}C\webright ) \]
making the diagrams
and
commute, then the natural transformation $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.
-
If there exists a natural transformation
\[ \alpha ^{\boxtimes ,\otimes }_{A,B,C}\colon \webleft (A\boxtimes _{\mathcal{C}}B\webright )\otimes _{\mathcal{C}}C\to A\otimes _{\mathcal{C}}\webleft (B\boxtimes _{\mathcal{C}}C\webright ) \]
making the diagrams
and
commute, then the natural transformation $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.