?? so let x be a category and y a full subcategory of _presheaf(x)_ with a left adjoint to its full embedding ...

??given a representable presheaf "[-,x1]" on x and a sub-presheaf s of it ... ??apply the left adjoint to the inclusion morphism ... ??and then consider the image inclusion of that ??? ??or something ???....

??maybe this doesn't generally give an endomorphism of the subobject classifier ???... ???...

??maybe it doesn't even come close .... ???....