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.