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?