Example 2.1.6.5 (Strict Monoidal Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be strict monoidal categories (Definition 2.1.2.1). We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strict monoidal if it is a nonunital strict monoidal functor (Definition 2.1.4.1) which carries the strict unit object $\mathbf{1}_{\operatorname{\mathcal{C}}}$ to the strict unit object $\mathbf{1}_{\operatorname{\mathcal{D}}}$.
Every strict monoidal functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ can be regarded as a monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, by taking each tensor constraint $\mu _{X,Y}$ to be the identity morphisms from $F(X) \otimes F(Y) = F(X \otimes Y)$ to itself. Conversely, if $(F, \mu )$ is a monoidal functor for which the tensor constraints $\mu _{X,Y}$ and the unit morphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$ are identity morphisms in $\operatorname{\mathcal{D}}$, then $F$ is a strict monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.