# Differential geometry – question about distributors with limitation

This is easy but has difficulty understanding a point. Prove that when $$f: X rightarrow Y$$ is thus a diffeomorphism of manifolds with boundary $$partially f$$ cards $$partial X$$ diffeomorphic too $$partial Y$$,

Answer: Leave $$U subset H ^ k$$ be an open subset and let $$phi: U rightarrow X$$ a parameterization of $$X$$, Then $$f circ phi: U rightarrow Y$$ is a parameterization of $$Y$$, Then $$partial Y cap f circ phi (U) = f circ phi ( partial U)$$so $$partial Y subset f ( partial X)$$ as $$Y$$ is covered by such parameterizations. Similar, $$partial X subset f ^ {- 1} ( partial Y)$$ and thus $$f ( partialX) = partial Y$$, I do not understand why $$partial Y cap f circ phi (U) = f circ phi ( partial U)$$, I understand $$f circ phi$$ is a diffeomorphism, but does not understand why he maps the boundary of $$U$$ to the border of $$Y$$, Thanks and thanks for a hint.