Pardon if this is an duplicate question. Say, for instance that I an embedded submanifold $N$ of a manifold $M$. I also know that $M$ is isomorphic (diffeomorphic?) to $M'$ via the isomorphism $f$. The embedded submanifold $N$ is defined via the regular level set theorem (see for instance Lee, ISM first edition, Corr. 8.10). Can I say that $N$ also is embedded in $M'$?
My attempt: We know that there is an immersion $F:N \to M$, and that this immersion is a topological embedding. It feels as if we could consider $f \circ F : N \to M' $, and while I see that this is an immersion, I fail to see that it is also a topological embedding. Is this not the way to go? Are there other things I can say?