Definition 2.1.4.3 (Nonunital Lax Monoidal Functors). 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 from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. A nonunital lax monoidal structure on $F$ is a collection of morphisms $\mu = \{ \mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y) \} _{X,Y \in \operatorname{\mathcal{C}}}$ 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
\[ \xymatrix@R =50pt@C=50pt{ F(X) \otimes F(Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ F(f) \otimes F(g) } & F( X \otimes Y) \ar [d]^{ F( f \otimes g) } \\ F(X') \otimes F(Y' ) \ar [r]^-{ \mu _{X',Y'} } & F( X' \otimes Y' ) } \]commutes (in the category $\operatorname{\mathcal{D}}$). In other words, we can regard $\mu $ as a natural transformation of functors as indicated in the diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\ar [r]^-{\otimes } \ar [d]_{F \times F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\times \operatorname{\mathcal{D}}\ar [r]^-{ \otimes } \ar@ {=>}[]+<20pt,20pt>;+<60pt,50pt>_-{\mu } & \operatorname{\mathcal{D}}. } \]- $(b)$
The morphisms $\mu _{X,Y}$ are compatible with the associativity constraints on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ in the following sense: for every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram
\[ \xymatrix@R =40pt@C=70pt{ F(X) \otimes (F(Y) \otimes F(Z) ) \ar [r]^-{ \alpha _{F(X), F(Y), F(Z) } } \ar [d]^{ \operatorname{id}_{F(X)} \otimes \mu _{Y,Z} } & (F(X) \otimes F(Y) ) \otimes F(Z) \ar [d]^{ \mu _{X,Y} \otimes \operatorname{id}_{F(Z)}} \\ F(X) \otimes F(Y \otimes Z) \ar [d]^{ \mu _{X,Y \otimes Z} } & F(X \otimes Y) \otimes F(Z) \ar [d]^{ \mu _{X \otimes Y, Z}} \\ F( X \otimes (Y \otimes Z) ) \ar [r]^-{ F( \alpha _{X,Y,Z } )} & F( (X \otimes Y) \otimes Z) } \]
commutes (in the category $\operatorname{\mathcal{D}}$).
A nonunital lax monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a pair $(F, \mu )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor and $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ is a nonunital lax monoidal structure on $F$. In this case, we will refer to the morphisms $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ as the tensor constraints of $F$.