Let $M^n$ be a smooth $n$-dimensional manifold and let $\mathbb{D}^n$ be the open unit disk in $\mathbb{R}^n$. Consider the space $\text{Emb}(\mathbb{D}^n,M^n)$ of smooth embeddings of $\mathbb{D}^n$ into $M^n$, equipped with the $C^{\infty}$ topology. Letting $\text{Fr}(M^n)$ be the bundle of frames of the tangent bundle of $M^n$, there is a map $\Psi\colon \text{Emb}(\mathbb{D}^n,M^n) \rightarrow \text{Fr}(M^n)$ taking an embedding $f\colon \mathbb{D}^n \rightarrow M^n$ to the point $f(0) \in M^n$ equipped with the image of the standard framing of $T_0 \mathbb{D}^n = \mathbb{R}^n$ under the derivative $D_0 f\colon T_0 \mathbb{D}^n \rightarrow T_{f(0)} M^n$.
For a paper I'm writing, I'm looking for a reference proving that $\Psi$ is a homotopy equivalence. I know several proofs of this, but all of them have slightly annoying aspects (at least if you write out all the details) so I would prefer to just give a reference.
