Kerodon

Remark 2.1.5.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor. The condition that $F$ is a lax monoidal functor depends only on the underlying nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, and not on the particular choice of units $(\mathbf{1}_{\operatorname{\mathcal{C}}}, \upsilon _{\operatorname{\mathcal{C}}})$ and $(\mathbf{1}_{\operatorname{\mathcal{D}}}, \upsilon _{\operatorname{\mathcal{D}}})$ for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively (see Remark 2.1.2.11).