ScalingStacks

Proof. The first isomorphism follows from the fact that $N$ preserves products and that $N([m])\approx F(m)$. The second isomorphism may be derived from the fact that $\iso(D^{I[n]})\approx(\iso D)^{I[n]}\approx(\iso D)^{[n]}$ for any category $D$, and thus in particular when $D=C^{[m]}$. ∎