ScalingStacks

Proof. The $\oo$-category $\disk_{n/M}^{B}$ is evidently nonempty, as it contains the object $(\emptyset\hookrightarrow M)$. We must then prove that the diagonal functor $\disk_{n/M}^{B}\to\disk_{n/M}^{B}\times\disk_{n/M}^{B}$ is final. This diagonal functor fits into a diagram among $\infty$-categories

that we now explain. The upper left $\infty$-category is that of Definition 3.20 applied to the fold map $\nabla\colon M\sqcup M\to M$; it is equipped with the indicated projection functors. The right vertical arrow is induced by the symmetric monoidal structure on $\mfld_{n}^{B}$, which is disjoint union. This right vertical arrow is an equivalence; an inverse is given by declaring its projection to each factor to be given by intersecting with the corresponding cofactor of the disjoint union. Therefore, to prove that the diagonal functor is final it is sufficient to prove that both of the projection functors ${\sf ev}_{1}$ and ${\sf ev}_{0}$ are final. The finality of ${\sf ev}_{0}$ is Lemma 3.21.

We explain that ${\sf ev}_{1}$ is final. Note that the functor $\nabla^{-1}\colon\disk_{n/M}^{B}\to\mfld_{n/M\sqcup M}^{B}$ factors through the full $\infty$-subcategory $\disk_{n/M\sqcup M}^{B}$. As so, there is a canonical identification between $\infty$-categories

$$\disk_{\nabla}~\simeq~\disk_{n/M}^{B}\underset{\disk_{n/M\sqcup M}^{B}}{\times}{\sf Ar}(\disk_{n/M\sqcup M}^{B})$$

over $\disk_{n/M}^{B}$. Through this identification, the composite functor

$$\disk_{n/M}^{B}\xrightarrow{~\nabla~}\disk_{n/M\sqcup M}^{B}\xrightarrow{~\sf const~}{\sf Ar}(\disk_{n/M\sqcup M}^{B})$$

determines a right adjoint to the functor ${\sf ev}_{1}$. The finality of ${\sf ev}_{1}$ thereby follows.

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