We know that for an immersion $j:U \to X$ the restriction functor $j^\*:{\cal O}\_X-mod \to {\cal O}\_u-mod$ 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 $j^\*:Qco(X) \to Qco(u)$.



left adjoint to restriction functor
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.


