ScalingStacks

Proof. Recognize $\mathcal{E}{\sf uc}_{n}\subset\disk_{n}$ as the full $\infty$-subcategory consisting of the connected $n$-manifolds. Specialize the second statement of Lemma 2.12 to $i=1$ to obtain an identification

$$M~\simeq~\mathcal{E}{\sf uc}_{n/M}~\simeq~\Emb(\mathbb{R}^{n},M)_{{\sf Top}(n)}\longrightarrow\BTop(n)$$

involving the homotopy ${\sf Top}(n)$-coinvariants. Manifestly, this map of spaces agrees with the tangent classifier in the case $M=\mathbb{R}^{n}$. By construction, this map is functorial in the argument $M\in\mfld_{n}$. The general case follows. ∎