Kerodon

Example 2.1.4.7. Let $M$ and $M'$ be nonunital monoids, regarded as nonunital monoidal categories having only identity morphisms (Example 2.1.1.3). Then nonunital lax monoidal functors from $M$ to $M'$ (in the sense of Definition 2.1.4.3) can be identified with nonunital monoid homomorphisms from $M$ to $M'$ (in the sense of Variant 1.3.2.8). Moreover, every nonunital lax monoidal functor from $M$ to $M'$ is automatically strict.