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Variant Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A colax monoidal structure on $F$ is a lax monoidal structure on the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$: that is, a collection of maps $\mu = \{ \mu _{X,Y}: F(X \otimes Y) \rightarrow F(X) \otimes F(Y) \} _{X,Y \in \operatorname{\mathcal{C}}}$ satisfying the requirements of Variant, together with the additional condition that there exists a counit $\epsilon : F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \rightarrow \mathbf{1}_{\operatorname{\mathcal{D}}}$ having the property that, for every object $X \in \operatorname{\mathcal{C}}$, the left and right unit constraints of $F(X)$ the inverses of the composite maps

\[ F(X) \xrightarrow { F(\lambda _ X)} F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes X) \xrightarrow { \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, X} } F( \mathbf{1}_{\operatorname{\mathcal{C}}}) \otimes F(X) \xrightarrow { \epsilon \otimes \operatorname{id}} \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F(X) \]
\[ F(X) \xrightarrow { F(\rho _ X)} F( X \otimes \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { \mu _{ X, \mathbf{1}_{\operatorname{\mathcal{C}}} } } F(X) \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { \operatorname{id}\otimes \epsilon } F(X) \otimes \mathbf{1}_{\operatorname{\mathcal{C}}}. \]