Strict monoidal category

A strict monoidal category is a monoidal category whose associator and left and right unitors are identities. Thus $(X\otimes Y)\otimes Z=X\otimes(Y\otimes Z)$ and $\mathbb1\otimes X=X=X\otimes\mathbb1$ hold as equalities of objects and functors, with the specified coherence maps equal to identity arrows.

Why strict notation is useful

Endofunctors of a category, with composition as tensor product and the identity functor as unit, form a strict monoidal category. Ordinary vector-space tensor products come with canonical associativity isomorphisms rather than literal equalities.

Every monoidal category is monoidally equivalent to a strict one. This permits coherent calculations without parentheses, but does not make the original category's associators literally identities. In particular, one should not discard associator data when specifying a particular fusion category.

References

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §§2.8–2.9; author-hosted text.