ScalingStacks

Proof. By A.2.2.5.4, since $\mathcal{C}_{\times}$ is symmetric monoidal, so is $\operatorname{Alg}_{\mathcal{D}}\left(\mathcal{C}\right)$ and, for every $X\in\mathcal{D}$, the evaluation functor $e_{X}\colon\operatorname{Alg}_{\mathcal{D}}\left(\mathcal{C}\right)\to\mathcal{C}_{\times}$ is a symmetric monoidal functor. On the underlying $\infty$-categories, $e_{X}$ also preserves finite products since it preserves all limits. Finally, we show that the collection of evaluation functors is jointly conservative since they can be presented as the composition of the conservative restriction functor

$$\operatorname{Alg}_{\mathcal{D}}\left(\mathcal{C}\right)\to\operatorname{Fun}\left(\mathcal{D},\mathcal{C}\right)$$

and the collection of evaluation functors

$$e_{X}\colon\operatorname{Fun}\left(\mathcal{D},\mathcal{C}\right)\to\mathcal{C},$$

which are jointly conservative by T.3.1.2.1. Now, by 3.2.4(1), $\operatorname{Alg}_{\mathcal{D}}\left(\mathcal{C}\right)$ is Cartesian. ∎