This finishes the proof.
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Proof of Item (a): Indeed, consider the diagram
whose boundary diagram is the diagram whose commutativity we wish to prove. Since:
- Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathcal{C}}$;
- Subdiagram $\webleft (2\webright )$ commutes trivially;
- Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\lambda ^{\mathcal{C}}$, where the equality $\rho ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}=\lambda ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}$ comes from
; - Subdiagram $\webleft (4\webright )$ commutes by the right monoidal unity of $\webleft (\text{id}_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}|\mathbb {1}}\webright )$;
so does the boundary diagram, and we are done.
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Proof of Item (b): Indeed, consider the diagram
whose boundary diagram is the diagram whose commutativity we wish to prove. Since:
- Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathcal{C}}$;
- Subdiagram $\webleft (2\webright )$ commutes trivially;
- Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\rho ^{\mathcal{C}}$, where the equality $\rho ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}=\lambda ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}$ comes from ;
- Subdiagram $\webleft (4\webright )$ commutes by the left monoidal unity of $\webleft (\text{id}_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}|\mathbb {1}}\webright )$;
so does the boundary diagram, and we are done.
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Proof of Item (c): Indeed, consider the diagram
Since:
- The boundary diagram commutes trivially;
- Subdiagram $\webleft (1\webright )$ commutes by Item (b);
it follows that the diagram
commutes. But since $\text{id}^{\otimes ,-1}_{\mathbb {1}_{\mathcal{C}},\mathbb {1}'_{\mathcal{C}}}$ is an isomorphism, it follows that the diagram $\webleft (\dagger \webright )$ also commutes, and we are done.
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Proof of Item (d): Indeed, consider the diagram
Since:
- The boundary diagram commutes trivially;
- Subdiagram $\webleft (1\webright )$ commutes by Item (a);
it follows that the diagram
commutes. But since $\text{id}^{\otimes ,-1}_{\mathbb {1}}$ is an isomorphism, it follows that the diagram $\webleft (\dagger \webright )$ also commutes, and we are done.
This finishes the proof.
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Proof of Item (a): We may partition the monoidality diagram for $\text{id}^{\otimes }$ of Item 2 of Remark 10.1.1.1.3 as follows:
Since:
- Subdiagram $\webleft (1\webright )$ commutes by Item (a) of Item 1.
- Subdiagram $\webleft (2\webright )$ commutes by assumption.
- Subdiagram $\webleft (3\webright )$ commutes by assumption.
it follows that the boundary diagram also commutes, i.e. $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.
-
Proof of Item (b): We may partition the monoidality diagram for $\text{id}^{\otimes }$ of Item 2 of Remark 10.1.1.1.3 as follows:
Since:
- Subdiagram $\webleft (1\webright )$ commutes by assumption.
- Subdiagram $\webleft (2\webright )$ commutes by assumption.
- Subdiagram $\webleft (3\webright )$ commutes by Item (b) of Item 1.
it follows that the boundary diagram also commutes, i.e. $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.
-
Proof of Item (c): We may partition the monoidality diagram for $\text{id}^{\otimes }$ of Item 2 of Remark 10.1.1.1.3 as follows:
Since subdiagrams (1) and (2) commute by assumption, it follows that the boundary diagram also commutes, i.e. $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.
This finishes the proof.
