ScalingStacks

Remark 9. The fact that $R_{A,-}$ is a pseudonatural equivalence can be expressed equivalently as follows: for any object $X\in\mathcal{C}$, there exists an equivalence $R_{A,X}\colon A\otimes X\to X\otimes A$, and for any morphism $f\colon X\to Y$ in $\mathcal{C}$, there exists a 2-isomorphism $R_{A,f}\colon(A\otimes f)\circ R_{A,Y}\Rightarrow R_{A,X}\circ(f\otimes A)$:

such that $(\bullet\otimes{\to}\to)$ and $(\bullet\otimes{\Downarrow})$ commute.

Similarly, the fact that $\>\tilde{R}_{(A|-,-)}$ is a modification means that the diagrams $(\bullet\otimes({\to}\otimes\bullet))$ and $(\bullet\otimes(\bullet\otimes{\to}))$ commute.