# Kerodon

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Example 2.1.4.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital strict monoidal functor. Then $F$ admits a nonunital monoidal structure $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$, where we take each $\mu _{X,Y}$ to be the identity morphism from $F(X) \otimes F(Y) = F(X \otimes Y)$ to itself.

Conversely, if $(F, \mu )$ is a nonunital monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ with the property that the tensor constraints $\mu _{X,Y}$ is an identity morphism in $\operatorname{\mathcal{D}}$, then $F$ is a nonunital strict monoidal functor.