ScalingStacks

Proof. Observe that for any $\mathcal{D}$ tensored over $\mathcal{C}$, the pullback

$$\Fun^{\rm L}_{\mathcal{C}}(\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M},\mathcal{D})\rightarrow\Fun^{\rm L}_{\mathcal{C}}(\mathcal{M},\mathcal{D})$$

induced by the induction $F:\mathcal{M}\to\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}$ is conservative. In other words, if a functor out of $\mathrm{Mod}_{A}\times\mathcal{M}$ (which preserves colimits in each variable) is trivial when restricted to $\mathcal{M}$, then it is necessarily trivial.

Consequently, switching to opposite categories, we have that the corresponding functor

$$\Fun^{\rm R}_{\mathcal{C}}(\mathcal{D},\mathrm{Mod}_{A}\otimes\mathcal{M})\rightarrow\Fun^{\rm R}_{\mathcal{C}}(\mathcal{D},\mathcal{M})$$

induced by the forgetful functor $G:\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}\to\mathcal{M}$ is conservative.

Now we can apply Lemma 4.2 with $\mathcal{M}^{\prime}=\mathrm{Mod}_{A}\otimes\mathcal{M}$ to obtain that $G$ is conservative. ∎