We know that for an immersion $j:U \to X$ the restriction functor $j^*:{\cal O}_Xmod \to {\cal O}_umod$ has a left adjoint $j!$. I am looking for some condition to deduce that $j!$ takes its values in Qco(X) that is to be a left adjoint for the functor $j^*:Qco(X) \to Qco(u)$.

The restriction functor $\mathrm{Qcoh}(X) \to \mathrm{Qcoh}(U)$ doesn't preserve infinite products in general (which always exist, by the way). Therefore it cannot have a left adjoint. 

