Local extendability of structure functor of A-germs of embeddings on a differential manifold

Question

I am reading "Differential Topology" by Morris W. Hirsch. In book some form of Whitney embedding theorem is almost proved (there are details left to verify): every differential manifold $M$ (paracompact, Hausdorff, with countable base) of dimension n can be embedded in $\mathbb R^{2n+1}$. It is achieved by technique of structure functors (I can't find anywhere else this technique, which leads me to the question where there could be more information about that, I am aware that language of sheaf theory is very similar). What is left to be done and seems mystery to me is the following: let $K_1, K_2, ...$ be a locally finite cover of $M$ by compact sets (we can assume each of them is a closed ball in some chart), let $W$ be some neighbourhood of (possibly infinite) sum $P$ of elements of this cover. Given a function $f: M \to \mathbb R^{2n+1}$ proper of a fixed differentiability class, s. t. $f_{|W}$ is embedding, and given a point $x$ of $M$, there are $K_n$ and proper function $g: M \to \mathbb R^{2n+1}$ (of the same differentiablity class) s. t. $x \in K_n$, $f_{|P}=g_{|P}$ and $g_{|A}$ is an embedding for $A$ being neighbourhood of $P \cup K_n$. Could someone help me with this? If $x$ is s. t. $K_n$ can be assumed to have empty intersection with $P$, it is easy, but in other case I don't know how to get injectivity.