In detail, the moduli category of monoidal structures on $\mathcal{C}$ is the category $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ where:
- Objects. The objects of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ are monoidal categories $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)$ whose underlying category is $\mathcal{C}$.
- Morphisms. A morphism from $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)$ to $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\webright)$ is a monoidal functor structure
\begin{gather*} \text{id}^{\otimes }_{\mathcal{C}} \colon A\otimes _{\mathcal{C}}B \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A\boxtimes _{\mathcal{C}}B,\\ \text{id}^{\otimes }_{\mathbb {1}|\mathcal{C}} \colon \mathbb {1}_{\mathcal{C}} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathbb {1}’_{\mathcal{C}} \end{gather*}
on the identity functor $\text{id}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ of $\mathcal{C}$.
- Identities. For each $M\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)\in \text{Obj}\webleft (\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )\webright )$, the unit map
\[ \mathbb {1}^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,M} \colon \text{pt}\to \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,M\webright ) \]
of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ at $M$ is defined by
\[ \text{id}^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\text{id}_{\otimes _{\mathcal{C}}},\text{id}_{\mathbb {1}_{\mathcal{C}}}\webright ), \]where $\Big(\text{id}_{\otimes _{\mathcal{C}}},\text{id}_{\mathbb {1}_{\mathcal{C}}}\Big)$ is the identity monoidal functor of
. - Composition. For each $M,N,P\in \text{Obj}\webleft (\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )\webright )$, the composition map
\[ \circ ^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,N,P}\colon \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (N,P\webright )\times \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,N\webright )\to \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,P\webright ) \]
of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ at $\webleft (M,N,P\webright )$ is defined by
\[ \Big(\text{id}^{\otimes ,\prime }_{\mathcal{C}},\text{id}^{\otimes ,\prime }_{\mathbb {1}|\mathcal{C}}\Big)\mathbin {\circ ^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,N,P}}\Big(\text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes }_{\mathbb {1}|\mathcal{C}}\Big)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Big(\text{id}^{\otimes ,\prime }_{\mathcal{C}}\circ \text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes ,\prime }_{\mathbb {1}|\mathcal{C}}\circ \text{id}^{\otimes }_{\mathbb {1}|\mathcal{C}}\Big). \]
