The Stacks project

Definition 4.43.9. A quadruple $(\mathcal{C}, \otimes , \phi , \psi )$ where $\mathcal{C}$ is a category, $\otimes : \mathcal{C} \otimes \mathcal{C} \to \mathcal{C}$ is a functor, $\phi $ is an associativity constraint, and $\psi $ is a commutativity constraint compatible with $\phi $ is called a symmetric monoidal category if there exists a unit.