Let us consider an embedding of smooth manifolds $i: (Y, \partial Y) \rightarrow (X, \partial X)$. It is neat (see Hirsh, "Differential topology") if $i(\partial Y) = i(Y) \cap \partial X$ and, for every $y \in \partial Y$, there exists a boundary chart $(U, \varphi)$ of $X$ in $i(y)$ such that $U \cap i(Y)$ is the counterimage via $\varphi$ of the half-space orthogonal to the boundary. In the paper of Hopkins and Singer "Quadratic functions in geometry, topology and M-theory", appendix C.2, they define a generic map $f: (Y, \partial Y) \rightarrow (X, \partial X)$ to be neat if $f^{-1}(\partial X) = \partial Y$ and, for every $y \in \partial Y$, the map $df_{y}: T_{y}Y/T_{y}\partial Y \rightarrow T_{f(y)}X/T_{f(y)}\partial X$ is an isomorphism. Then they state the following theorem, whose proof I cannot find in the references they give:
Theorem C.17 If $f: Y \rightarrow X$ is a neat map of compact t-manifolds (for me it's enough to consider manifolds with boundary), there exist $N >> 0$ and a factorization $X \hookrightarrow Y \times \mathbb{R}^{N} \rightarrow Y$ of $f$ through a neat embedding $X \hookrightarrow Y \times \mathbb{R}^{n}$.
My questions are:
1) Is it true that the embedding $X \hookrightarrow Y \times \mathbb{R}^{n}$ is stably unique up to isotopy, as it happens for manifolds without boundary, and for embeddings $X \hookrightarrow \mathbb{R}^{n}_{+}$ of manifolds with boundary? Where can I find a proof?
2) Given a generic map $f: (Y, \partial Y) \rightarrow (X, \partial X)$ between compact smooth manifolds, is it always homotopic to a neat map, maybe realtively to $\partial Y$?
