Existence of a right adjoint functor of the inverse image via a morphism of schemes between the categories of quasi-coherent modules

Question

Let $f:X\to Y$ a morphism of schemes. It induces a covariant functor: $$f^*:Qcoh(Y)\to Qcoh(X)$$ Which happens to be the inverse image. Now, fixing any quasi-coherent $O_X$-module N we can define another functor: $\require{AMScd}$ \begin{CD} Qcoh(Y) @>>> \mathcal{S}ets\\ \\ M@>>>Hom_{O_X}(f^*(M),N) \end{CD} which, due to representable functor theorem, it is representable. Then, $f^*$ has a right adjoint, and it has to be the direct image $f_*$.

The problem I find now is that the direct image of a quasi-coherent sheaf it is not necessarily quasi-coherent, so there is something wrong, but I can not find where the error is.