ScalingStacks

Remark 5.5. Unwinding the functor $\phi_{n}^{k}$ in the argument above, one finds that the product in the category $\corr_{n}^{k}$ may be written recursively in the following manner:

$$(X_{0},X_{1},F)\times(Y_{0},Y_{1},G)\cong(X_{0}\times Y_{0},X_{1}\times Y_{1},F\otimes G)$$

where $F\otimes G$ is defined as the composite:

$$(X_{0}\times Y_{0})^{\mathrm{op}}\times(X_{1}\times Y_{1})\cong(X_{0}^{\mathrm{op}}\times X_{1})\times(Y_{0}^{\mathrm{op}}\times Y_{1})\stackrel{{\scriptstyle(F,G)}}{{\longrightarrow}}\corr_{n-1}^{k-1}\times\corr_{n-1}^{k-1}\stackrel{{\scriptstyle\times}}{{\longrightarrow}}\corr_{n-1}^{k-1}.$$