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

Question

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?

Answer 1

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.