Reference for transversality -- relative version

Question

In Hirsch, Differential topology, Thm 2.1 states that for smooth manifolds $M,N$ and a closed submanifold $A\subseteq N$, the space of maps from $M$ to $N$ transverse to $A$ is dense and open in $C^\infty(M,N)$ (in the strong topology).

I'm looking for a reference (or simple reason) for the following relative version of this (if it holds): if $C\subseteq M$ is closed (submanifold, if necessary), $f: M\to N$ smooth and $f$ is transverse to $A$ on $C$, then the space $$ \{g\in C^\infty(M,N):\,\,\text{$g$ transverse to $A$ and $g|_C=f|_C$}\} $$ is also dense and open in $\{g\in C^\infty(M,N):\,\,\text{$g|_C=f|_C$}\}$. Any idea please?

Answer 1

The result is true. It is a consequence of the parametrized density of transversality, as one can see in Abraham-Robbin's Transversal mappings and flows. I believe that there there is also the relative version. In any case, this relative versión follows by adding a bump function to the standard construction that preserves $f$ in some nbhd of $C$. Note that here it is important that $A$ is closed in $N$: this implies that transversality on $C$ extends to transversality on some nbhd of $C$, and consequently you can keep the same $f$ on some nbhd of $C$.