Denote $\mathbf{LRS}$ the category of locally ringed spaces (where the stalks of morphisms are local morphisms), and $\mathbf{SRng}$ a small category of (commutative unital) rings. For $X\in\mathbf{LRS}$, there is a functor $S_X\colon\mathbf{SRng}\to\mathbf{Set}$ specified by $S_X(A)=\mathbf{LRS}(\operatorname{Spec}A,X)$, which induces a functor $S\colon\mathbf{LRS}\to\mathbf{Funct}(\mathbf{SRng},\mathbf{Set})$.
(Notation: Given a category $\mathcal C$, $\mathcal C(X,Y)$ is the set of morphisms between $X,Y\in\mathcal C$. We assume that all categories are locally small)
The theorem of existence of geometric realization, claims that $S$ has a left adjoint, called the geometric realization functor. In Demazure & Gabriel's Introduction to Algebraic Geometry and Algebraic Groups, they claim that it is a particular case of a well-known theorem of Kan. However, they sketch a proof for this special case.
I wonder what is the well-known theorem of Kan in question? The proof they sketch works along the line that every functor $\mathbf{SRng}\to\mathbf{Set}$ is a colimit of representable functors (via category of elements), and for representable functors $\mathbf{SRng}(A,-)$, they just define the geometric realization as $\operatorname{Spec}A$.
Let $\cal C$ be small and $\cal D$ be cocomplete, and let $\text{Spec}\colon \mathcal C \to \cal D$ be a functor. Then there is an adjunction $$ \text{Lan}_y\text{Spec} \dashv \text{Lan}_\text{Spec} y $$ between the left Kan extension of Spec along the yoneda embedding, and the left Kan extension of the yoneda embedding along Spec.
(the functor $\text{Lan}_\text{Spec} y$ coincides with $\hom(\text{Spec}(-),=)$)