In "introduction to algebraic geometry and algebraic groups" Gabriel and Demazure proved the following theorem(section 4 theorem 4.1.):

Let $\mathcal{LRS}\rightarrow{Sh}_{Zar}$ be the functor that acts on objects as follows:$$X\mapsto{\mathcal{LRS}(Spec(-),X)}$$ Then it has a left adjoint.

$\mathcal{LRS}$ denotes the category of locally ringed spaces and $Sh_{Zar}$ denotes the category of sheaves on commutative rings with respect to zariski topology.

Is this theorem true? Authors are using in proof some colimits formulas over large index category. The problem is that $\mathcal{LRS}$ is small cocomplete, so there are no reasons to believe that such colimit would be contained in chosen universe. Moreover, if this theorem holds, then one will always have coarse moduli space for any zariski sheaf at least as locally ringed space. So I really have doubts about it.