In particular, a morphism in $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ 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 $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\webright)$ satisfies the following conditions:
-
Naturality. For each pair $f\colon A\to X$ and $g\colon B\to Y$ of morphisms of $\mathcal{C}$, the diagram
commutes.
-
Monoidality. For each $A,B,C\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram
commutes.
-
Left Monoidal Unity. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram
commutes.
-
Right Monoidal Unity. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram
commutes.
