Q:

Equivalent definition of Schemes

I recall seeing that the category of schemes can be captured by a general construction as follows. 
Let $\mathbf{Spec}\colon \mathbf{CRing}^{op}\to \mathbf{LRS}$ be the usual functor from the category of commutative rings to the category of locally ringed spaces by assigning a ring to its structured sheaf. 
Now, this is where I am a bit hesitant to continue, since we are to take the Yoneda extension along $\mathbf y\colon \mathbf{CRing}^{op}\to \mathbf{Set}^{\mathbf{CRing}}$ to a essentially unique functor $L\colon \mathbf{Set}^{\mathbf{CRing}}\to \mathbf{LRS}$ such that $L\circ \mathbf y=\mathbf{Spec}$. By the general construction, $L$ has a right adjoint, 
$R\colon \mathbf{LRS}\to \mathbf{Set}^{\mathbf{CRing}}$.
My concern is that the usual construction of $L$ is taken to be $L(F)=colim_{el(F)}\mathbf{Spec}\circ \pi_F$, where $\pi_F\colon el(F)\to \mathbf{CRing}$ is the natural projection. However, $el(F)$ is not guaranteed to be small and $\mathbf{LRS}$ does not have all large colimits. 
Is there a way to make this idea work so that the category of schemes is equivalent to the full subcategory consisting of locally ringed spaces $(X,\mathscr O_X)$ such that the unit of the adjunction $\eta_X\colon LR(X)\to X$ is an isomorphism? Perhaps we should restrict $\mathbf{Set}^{\mathbf{CRing}}$ to be the smallest subcategory which includes only those objects which have colimits we want.

A:

Replace $\mathsf{Set}^{\mathsf{CRing}}$ by the category of functors $\mathsf{CRing} \to \mathsf{Set}$ which are cofinally small. This is the free cocompletion $\widehat{\mathsf{Ring}^{\mathrm{op}}}$. See here for the (sketch of) proof  that schemes have cofinally small functors of points. But I think that the subcategory of $\mathsf{LRS}$ of fixed points of the adjunction $\widehat{\mathsf{Ring}^{\mathrm{op}}} \leftrightarrow \mathsf{LRS}$  is strictly larger than $\mathsf{Sch}$.
If you want to have a functorial framework for schemes, just define them as functors $X : \mathsf{CRing} \to \mathsf{Set}$ which are sheaves (this can be written down using localizations in an elementary way) and locally representable (this takes some time). One can develop algebraic geometry that way, see for example Grothendieck's "Functorial Algebraic Geometry", Lecture notes written by Federico Gaeta.

