I'm trying to read Ščedrov's Forcing and Classifying Topoi, and there's a bit in 1.1 that is frequently references, but I don't quite understand its import. If I'm not missing the point, 1.1 is essentially a description of his overall method of engineering a category of sheaves to contain an object with certain properties.
There is then this paragraph:
Additional geometric requirements (expressed through possibly infinitary disjunctions of formulae) define a Grothendieck topology on $C^{op}$ (alternatively, a geometric theory $T_2$, cf. [MR], Chapter 9). This corresponds to "cutting down" in the examples at the beginning of this section, cf. [Jo 1, §3.4.]. $[\ldots]$ The object we need is then (as a universal model of $T_2$) a subobject of (the associated sheaf of) the "diagonal" functor $G$ in $S^C$, $G$ arising from the evaluation functor by the Yoneda embedding: $$G=\sum_{c'\in C}Hom_C(c',-),$$ the generic object in $S^C$.
(iv) Since $G$ is a universal object with the required geometric properties, it might satisfly further non-geometric requirements (e.g. preservation of cardinals in set-theoretic focing).
Why is $G$ the object of interest in Ščedrov's scheme, and what is its universal property? This appears to be the final step in his outline of his approach, but I am having a hard time understanding the motivation for the details here; but it's clearly important as he uses various forms of $G$ in later chapters.
(In case it's relevant to the reader, [MR] is Makkai & Reyes's First Order Categorical Logic, and [Jo 1] is Johnstone's Topos Theory.)