Proof of Proposition 2.1.4.21.
Unwinding the definitions, we see that to describe nonunital monoidal structure on the category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ satisfying condition $(1)$, one must give the following data:
For every pair of objects $(C, D, \eta )$ and $(C', D', \eta ')$ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, we must supply a tensor product $(C, D, \eta ) \otimes (C', D', \eta ' )$. By virtue of the assumption that $U$ is nonunital strict monoidal, this tensor product must be given as a triple $(C \otimes C', D \otimes D', t(\eta , \eta ') )$, for some morphism $t(\eta ,\eta '): C \otimes C' \rightarrow G(D \otimes D')$ in the category $\operatorname{\mathcal{D}}$.
For every pair of morphisms $(u,v): (C,D, \eta ) \rightarrow (\overline{C}, \overline{D}, \overline{\eta } )$ and $(u',v'): (C', D', \eta ') \rightarrow (\overline{C}', \overline{D}', \overline{\eta }')$ in the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, we must supply a tensor product morphism $( C \otimes C', D \otimes D', t(\eta ,\eta ') ) \rightarrow ( \overline{C} \otimes \overline{C}', \overline{D} \otimes \overline{D}', t( \overline{\eta }, \overline{\eta }' ) )$. Note that this morphism is uniquely determined: for $U$ to be a nonunital strict monoidal functor, it must be the pair $(u \otimes u', v \otimes v')$. However, the existence of this morphism imposes the following condition:
- $(i)$
If the diagrams
\[ \xymatrix@R =50pt@C=50pt{ C \ar [r]^-{\eta } \ar [d]^{u} & G(D) \ar [d]^{ G(v) } & C' \ar [r]^-{\eta '} \ar [d]^{u'} & G(D') \ar [d]^{ G(v') } \\ \overline{C} \ar [r]^-{ \overline{\eta } } & G( \overline{D} ) & \overline{C}' \ar [r]^-{ \overline{\eta }' } & G( \overline{D}' ) } \]
commute (in the category $\operatorname{\mathcal{C}}$), then the diagram
\[ \xymatrix@C =50pt@R=50pt{ C \otimes C' \ar [r]^-{ t(\eta ,\eta ') } \ar [d]^{u \otimes u'} & G(D \otimes D') \ar [d]^{ G( v \otimes v') } \\ \overline{C} \otimes \overline{C}' \ar [r]^-{ t( \overline{\eta },\overline{\eta }') } & G( \overline{D} \otimes \overline{D}' ) } \]
also commutes.
For every triple of objects $(C, D, \eta )$, $(C', D', \eta ')$, and $(C'', D'', \eta '' )$ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, we must supply an associativity constraint
\[ (C,D, \eta ) \otimes ( ( C', D', \eta ') \otimes (C'', D'', \eta '') ) \simeq ( (C,D, \eta ) \otimes (C', D', \eta ') ) \otimes (C'', D'', \eta '' ) \]
in $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. By virtue of our assumption that $U$ is nonunital strict monoidal, this associativity constraint is uniquely determined: it must be the pair $(\alpha _{C,C',C''}, \alpha _{D,D',D''} )$ given by the associativity constraints for the nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. However, the existence of this morphism imposes the following condition:
- $(ii)$
For every triple of morphisms $\eta : C \rightarrow G(D)$, $\eta ': C' \rightarrow G(D')$, and $\eta '': C'' \rightarrow G(D'')$, the diagram
\[ \xymatrix@C =50pt@R=50pt{ C \otimes (C' \otimes C'') \ar [r]^-{ \alpha _{C,C',C''} } \ar [d]^-{ t(\eta , t(\eta ',\eta '') )} & (C \otimes C') \otimes C'' \ar [d]^-{ t( t(\eta , \eta '), \eta '' )} \\ G(D \otimes (D' \otimes D'') ) \ar [r]^-{ G( \alpha _{D,D',D''} ) } & G( (D \otimes D') \otimes D'' ) } \]
commutes (in the category $\operatorname{\mathcal{C}}$).
If this condition is satisfied, then the associativity constraints are automatically functorial and satisfy the pentagon identity (since the analogous conditions hold in the categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively).
Given a collection of morphisms $t(\eta , \eta ')$ satisfying these conditions, we define $\mu = \{ \mu _{D,D'} \} _{D,D' \in \operatorname{\mathcal{D}}}$ by the formula $\mu _{D,D'} = t( \operatorname{id}_{G(D)}, \operatorname{id}_{ G(D') })$. Note that, if $(C,D, \eta )$ and $(C', D', \eta ')$ are arbitrary objects of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, then we have canonical maps
\[ (\eta , \operatorname{id}_ D): (C,D,\eta ) \rightarrow (G(D), D, \operatorname{id}_{G(D)}) \quad \quad (\eta ', \operatorname{id}_{D'} ): (C',D', \eta ') \rightarrow (G(D'), D', \operatorname{id}_{ G(D') } ). \]
Applying condition $(i)$, we see that the morphism $t(\eta ,\eta ')$ can then be recovered as the composition
\[ C \otimes C' \xrightarrow {\eta \otimes \eta '} G(D) \otimes G(D') \xrightarrow { \mu _{D,D'} } G(D \otimes D'). \]
To complete the proof, it will suffice to show that if we are given any system of morphisms $\mu = \{ \mu _{D,D'}: G(D) \otimes G(D') \rightarrow G(D \otimes D') \} _{D,D' \in \operatorname{\mathcal{D}}}$ and we define $t(\eta ,\eta ')$ as above, then $\mu $ is a nonunital lax monoidal structure on $G$ if and only if conditions $(i)$ and $(ii)$ are satisfied.
Using the formula for $t(\eta , \eta ')$ in terms of $\mu $, we can rewrite condition $(i)$ as follows:
- $(i')$
If the diagrams
\[ \xymatrix@R =50pt@C=50pt{ C \ar [r]^-{\eta } \ar [d]^{u} & G(D) \ar [d]^{ G(v) } & C' \ar [r]^-{\eta '} \ar [d]^{u'} & G(D') \ar [d]^{ G(v') } \\ \overline{C} \ar [r]^-{ \overline{\eta } } & G( \overline{D} ) & \overline{C}' \ar [r]^-{ \overline{\eta }' } & G( \overline{D}' ) } \]
commute (in the category $\operatorname{\mathcal{C}}$), then the outer rectangle in the diagram
\[ \xymatrix@C =50pt@R=50pt{ C \otimes C' \ar [r]^-{\eta \otimes \eta '} \ar [d]^{u \otimes u'} & G(D) \otimes G(D') \ar [r]^-{ \mu _{D,D'} } \ar [d]^{ G(v) \otimes G(v')} & G(D \otimes D') \ar [d]^{G(v \otimes v')} \\ \overline{C} \otimes \overline{C}' \ar [r]^-{\overline{\eta } \otimes \overline{\eta }'} & G( \overline{D} ) \otimes G( \overline{D}' ) \ar [r]^-{ \mu _{ \overline{D}, \overline{D}' } } & G( \overline{D} \otimes \overline{D}' ) } \]
commutes.
Note that the left square appearing in this diagram is automatically commutative. Assertion $(i')$ is therefore a consequence of the following:
- $(a)$
For every pair of morphisms $v: D \rightarrow \overline{D}$ and $v': D' \rightarrow \overline{D}'$ in the category $\operatorname{\mathcal{D}}$, the diagram
\[ \xymatrix@C =50pt@R=50pt{ G(D) \otimes G(D') \ar [r]^-{ \mu _{D,D'} } \ar [d]^{ G(v) \otimes G(v') } & G(D \otimes D') \ar [d]^{G( v \otimes v')} \\ G( \overline{D} ) \otimes G( \overline{D}' ) \ar [r]^-{ \mu _{ \overline{D}, \overline{D}' } } & G( \overline{D} \otimes \overline{D}' ) } \]
commutes (in the category $\operatorname{\mathcal{C}}$).
Conversely, if $(i')$ is satisfied, then $(a)$ can be deduced by specializing to the case $\eta = \operatorname{id}_{G(D)}$, $\eta ' = \operatorname{id}_{G(D')}$, $\overline{\eta } = \operatorname{id}_{ G(\overline{D})}$, and $\overline{\eta }' = \operatorname{id}_{ G( \overline{D}')}$. It follows that $(i)$ is satisfied if and only if $(a)$ is satisfied: that is, if and only if $\mu = \{ \mu _{D,D'} \} _{D,D' \in \operatorname{\mathcal{D}}}$ is a natural transformation.
We can reformulate condition $(ii)$ as follows:
- $(ii')$
For every triple of morphisms $\eta : C \rightarrow G(D)$, $\eta ': C' \rightarrow G(D')$, and $\eta '': C'' \rightarrow G(D'')$, the outer rectangle in the diagram
\[ \xymatrix@C =80pt@R=50pt{ C \otimes (C' \otimes C'') \ar [r]^-{ \alpha _{C,C',C''} } \ar [d]^{\eta \otimes (\eta ' \otimes \eta '') } & (C \otimes C') \otimes C'' \ar [d]^{( \eta \otimes \eta ') \otimes \eta ''} \\ G(D) \otimes (G(D') \otimes G(D'') ) \ar [r]^-{ \alpha _{ G(D), G(D'), G(D'')}} \ar [d]^{\operatorname{id}_{G(D)} \otimes \mu _{D,D'} } & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{\mu _{D,D'} \otimes \operatorname{id}_{G(D'')} } \\ G(D) \otimes G(D' \otimes D'') \ar [d]^{\mu _{D,D'\otimes D''}} & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{ \mu _{ D \otimes D', D''} } \\ G(D \otimes (D' \otimes D'')) \ar [r]^-{ G( \alpha _{D,D',D''} ) } & G( (D \otimes D') \otimes D'') } \]
commutes (in the category $\operatorname{\mathcal{C}}$).
Since the upper square in this diagram automatically commutes (by the naturality of the associativity constraints on $\operatorname{\mathcal{C}}$), assertion $(ii')$ is a consequence of the following simpler assertion:
- $(b)$
For every triple of objects $D,D',D'' \in \operatorname{\mathcal{D}}$, the diagram
\[ \xymatrix@C =80pt{ G(D) \otimes (G(D') \otimes G(D'') ) \ar [r]^-{ \alpha _{ G(D), G(D'), G(D'')}} \ar [d]^{\operatorname{id}_{G(D)} \otimes \mu _{D,D'} } & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{\mu _{D,D'} \otimes \operatorname{id}_{G(D'')} } \\ G(D) \otimes G(D' \otimes D'') \ar [d]^{\mu _{D,D'\otimes D''}} & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{ \mu _{ D \otimes D', D''} } \\ G(D \otimes (D' \otimes D'')) \ar [r]^-{ G( \alpha _{D,D',D''} ) } & G( (D \otimes D') \otimes D'') } \]
commutes (in the category $\operatorname{\mathcal{C}}$).
Conversely, if $(ii')$ is satisfied, then $(b)$ can be deduced by specializing to the case $\eta = \operatorname{id}_{G(D)}$, $\eta ' = \operatorname{id}_{ G(D')}$, and $\eta '' = \operatorname{id}_{ G(D'')}$. We conclude by observing that conditions $(a)$ and $(b)$ assert precisely that $\mu $ is a nonunital lax monoidal structure (Definition 2.1.4.3).
$\square$