Well, the exercise is the following:
If $f: X \to Y$ diffeomorphism and $X,Y$ manifolds with boundary then $f(\partial X) = \partial Y$
My idea to prove this is the following:
By the definition of $\partial X$ I use that if $x \in \partial X$ then $\exists \ \phi : U \subset H^k \to U' \subset X $ with $\phi$ a diffeomorphism. Also, $x \in U' \cap \partial X = \partial U'$ and $\phi^{-1}(x) = u \in \partial U$. So as I know that $f$ is also a diffeomorophism I used that $f(U') = V$ with $f(x) = y \in V \subset Y$. And also $f : U' \to V$ is a diffeomorphism. So if I compone $f \circ \phi : U \subset H^k \to V \subset Y$ we can see that $f \circ \phi (u) = y \in V \subset Y$. So by definition of boundary of a manifold we have that $f(x) = y \in \partial V \subset \partial Y $. So right now I have prove that $f(\partial X) \subset \partial Y$. To prove $\partial Y \subset f(\partial X)$ we prove $f^{-1}(\partial Y) \subset \partial X$ as we have prove $f(\partial X) \subset \partial Y$.
I don't know if my idea is correct or not.