Kerodon

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Variant 2.1.4.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A nonunital colax monoidal structure on $F$ is a nonunital lax monoidal structure on the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ (Definition 2.1.4.3). In other words, a colax monoidal structure on $F$ is a collection of morphisms $\mu = \{ \mu _{X,Y}: F(X \otimes Y) \rightarrow F(X) \otimes F(Y) \} _{X,Y \in \operatorname{\mathcal{C}}}$ which satisfy the following pair of conditions:

$(a)$

The morphisms $\mu _{X,Y}$ depend functorially on $X$ and $Y$: that is, for every pair of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$ in $\operatorname{\mathcal{C}}$, the diagram

\[ \xymatrix { F(X \otimes Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ F(f \otimes g) } & F( X ) \otimes F(Y) \ar [d]^{ F( f) \otimes F(g) } \\ F(X' \otimes Y' ) \ar [r]^-{ \mu _{X',Y'} } & F( X') \otimes F(Y' ) } \]

commutes (in the category $\operatorname{\mathcal{D}}$).

$(b)$

For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram

\[ \xymatrix@R =40pt@C=70pt{ F(X \otimes (Y \otimes Z) ) \ar [r]^-{ F(\alpha _{X,Y,Z}) } \ar [d]^{ \mu _{X, Y \otimes Z} } & F( (X \otimes Y) \otimes Z) \ar [d]^{ \mu _{X \otimes Y,Z }} \\ F(X) \otimes F(Y \otimes Z) \ar [d]^{ \operatorname{id}\otimes \mu _{Y,Z} } & F(X \otimes Y) \otimes F(Z) \ar [d]^{ \mu _{X,Y} \otimes \operatorname{id}} \\ F( X) \otimes (F(Y) \otimes F(Z)) \ar [r]^-{ \alpha _{F(X),F(Y),F(Z) } } & (F(X) \otimes F(Y)) \otimes F(Z)} \]

commutes.