ScalingStacks

Proof. Through the defining adjunctions for over $\infty$-categories, the first assertion follows because, for each $\infty$-category $\mathcal{K}$, the canonical diagram among $\infty$-categories

is a pushout; here, for $\mathcal{K}$ and $\mathcal{J}$ $\infty$-categories,

$$\mathcal{K}\star\mathcal{I}~:=~\mathcal{K}\underset{\mathcal{K}\times\{0\}\times\mathcal{I}}{\coprod}\mathcal{K}\times\{0<1\}\times\mathcal{I}\underset{\mathcal{K}\times\{1\}\times\mathcal{I}}{\coprod}\mathcal{I}$$

denotes the join of $\infty$-categories. The second assertion follows directly from the universal property of coproducts. The final assertion follows from Proposition 3.2.2.1 of [Lu2], which in particular gives that, for each Cartesian closed presentable $\infty$-category $\mathcal{C}$, the forgetful functor from commutative algebras $\Alg_{\sf Com}(\mathcal{C}^{\times})\to\mathcal{C}$ preserves and creates limits. Apply this result to the case $\mathcal{C}=\Cat$.

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