We know the isomorphism theorem states that for a Group G and subgroup H and normal subgroup N of it, We have : $H/H$ $\cap$ $N \cong HN/N$ There is a Ambiguity for me that in the proof of this statement we use map $\phi : H \to HN/N$ with $h\to hN$ and using first isomorphism theorem to get the statement. but with this reasoning we can also use $H/N$ instead of $HN/N$.
However why we can't say $H/N$ is just $HN/N$?