In the context of this question about the projection formula for sheaves, Giulio Bresciani answers and uses that the natural morphism $\gamma \colon \mathcal E\to f_*f^*\mathcal E$ is an isomorphism when $\mathcal E$ is free. As I said in my comment I don't understand why this is true, considering for example $\mathcal E=\mathcal O_Y$ we would get $\mathcal O_Y\cong f_*\mathcal O_X$ which is not always true.
So is it true and under what assumptions do we have an isomorphism (except $f$ being a fibre) ?