Kerodon

Warning 2.1.4.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories. A nonunital strict monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ possessing certain properties. However, a nonunital (lax) monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ together with additional structure, given by the tensor constraints $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y)$. We will often abuse terminology by identifying a nonunital (lax) monoidal functor $(F, \mu )$ with the underlying functor $F$; in this case, we implicitly assume that the tensor constraints $\mu _{X,Y}$ have been specified.