ScalingStacks

Let \(\mathcal A\) and \(\mathcal B\) be monoidal \(1\)-categories, with monoidal product denoted by \(\boxtimes\) in both cases and with associators \(b_{x,y,z}\) in \(\mathcal B\). A prebraiding \(\beta\) on a monoidal functor \(F\colon \mathcal A\rightarrow\mathcal B\) consists of the data of isomorphisms \[F(x)\boxtimes F(y) \xrightarrow{\beta_{x,y}} F(y) \boxtimes F(x)\qquad \forall x,y\in \mathcal A\] that form a natural transformation \(\boxtimes\circ (F\times F) \Rightarrow \boxtimes^{\mathrm{op}}\circ (F\times F)\) and satisfy the following two hexagon axioms for all \(x,y,z\in \mathcal A\): where the isomorphisms \(\simeq\) are part of the data of \(F\). (Supressing them provides the hexagon shapes.)