Definition 4.43.1. A triple $(\mathcal{C}, \otimes , \phi )$ where $\mathcal{C}$ is a category, $\otimes : \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a functor, and $\phi $ is an associativity constraint is called a *monoidal category* if there exists a unit $\mathbf{1}$.

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