Kerodon

Warning 2.1.6.3. 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. If $F$ is a monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, then it is both a nonunital monoidal functor (that is, the tensor constraints $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F( X \otimes Y)$ are isomorphisms) and a lax monoidal functor (that is, it admits a unit $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$). However, the converse is false: to qualify as a monoidal functor, $F$ must satisfy the additional condition that $\epsilon $ is an isomorphism.