ScalingStacks

the 1-groupoid of weak monoidal factorizations of \(F\) through \(\mathrm{ev}_{1_{\mathcal A}} \colon Z(F) \rightarrow\mathcal B\), i.e. the groupoid whose objects are pairs \((s,\eta)\) of a monoidal functor \(s \colon \mathcal A\rightarrow Z(F)\) and a monoidal natural isomorphism \(\eta \colon \mathrm{ev}_{1_{\mathcal A}} \circ s \Rightarrow F\) and whose morphisms \((s, \eta) \rightarrow(s', \eta')\) are monoidal natural isomorphisms \(\mu \colon s\Rightarrow s'\) such that \[\left( \mathrm{ev}_{1_{\mathcal A}} \circ s \xRightarrow{\mathrm{ev}_{1_{\mathcal A}} \circ \mu} \mathrm{ev}_{1_{\mathcal A}} \circ s' \xRightarrow{\eta'} F \right)=\left( \mathrm{ev}_{1_{\mathcal A}} \circ s \xRightarrow{\eta}F\right).\]