Pullback of the direct image of a vector bundle surjects to the vector bundle

Let $$f:X\rightarrow Y$$ be a finite morphism of smooth projective varieties over the field of complex numbers. Let $$E$$ be a locally free sheaf over $$X$$. We have the natural morphism $$\phi:f^*f_*E\rightarrow E$$. Does the finiteness of $$f$$ imply that $$\phi$$ is surjective? If $$f$$ is a closed immersion, then the above morphism is an isomorphism. If it is an arbitrary finite map, then is $$\phi$$ onto?

Since $$f$$ is finite, the functor $$f_*$$ is exact and conservative, so surjectivity of $$f^*f_*E \to E$$ is equivalent to surjectivity of $$f_*f^*f_*E \to f_*E.$$ On the other hand, by adjunction there is also a natural morphism $$f_*E \to f_*f^*f_*E$$ and the composition $$f_*E \to f_*f^*f_*E \to f_*E$$ is the identity. This proves required surjectivity.
• thank you for the answer. Why is the direct image functor $f_*$ conservative when $f$ is finite. Commented Feb 4, 2019 at 9:24
• Since $f_*$ is exact, for conservativity it is enough to note that $f_*F = 0$ implies $F = 0$. Commented Feb 4, 2019 at 9:34