Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to satisfy this functor in order to "extend" to a functor $F^*$ form the category of $\operatorname{Spec}(L)$-Schemes to the category of $\operatorname{Spec}(K)$-Schemes?

It is clear that the extension, if it exists, is unique, by Yoneda's lemma.

On the other hand, I am not sure any functor will work; it seems to me we need that the functor sends "open affine immersions" to "open affine immersions", i.e. flat morphisms $g:R\to S$ of finite presentation which are epimorphism, to the same type of morphisms, in order to be able to "patch" the image of an affine covering of a $\operatorname{Spec}(L)$-scheme to get a $\operatorname{Spec}(K)$-scheme. But I don't see how to do it if the intersections are not affine, and one needs to get an affine covering of the intersections, and so on. Moreover, I don't know if it is sufficient to show this for principal open schemes, so for maps of the type $g:R\to R[1/a]$ for $a\in R$.

I am interested, for example, in the functor sending any ring $R$ to its total ring of fractions $Q(R)$. I have not seen this mentioned in the scheme world. But such a general result, if it exists in the literature, could be useful to get e.g. the base change functor, or the reduced subcheme functor, etc.

necessarythat the functor preserve open embeddings or open coverings or intersections – after all you could start with an arbitrarily bad functor of schemes that happens to preserve affine schemes. $\endgroup$requirethat the extension has some properties? (If not, the question is a bit meaningless.) Also, I am very confused about the last paragraph, there is a well-known sheaf $K_X$ of meromorphic functions for any scheme $X$. $\endgroup$