The canonical map from a scheme to its fiber square over the target.
Let f:Y→X be a morphism of schemes. The two identity maps from Y to itself have the same composite f to X. The universal property of the fiber product therefore gives a unique morphism
Δf:Y⟶Y×XY
whose composites with both projections Y×XY→Y are the identity. This is the diagonal morphism of f.
The notation is the scheme-theoretic analogue of the point-set diagonal y↦(y,y). Its scheme structure also records infinitesimal information about the fibers of f, which is why properties of the diagonal characterize separation and ramification conditions.
In the Galois example, SpecK×SpecFSpecK records two copies of SpecK constrained to lie over the same base. It is the scheme-theoretic version of “pairs in one fiber.”
Given morphisms of schemesX→S←Y, their fiber product is a scheme X×SY with projections to X and Y whose composites to S agree, and which is universal with that property: for every scheme T,
Hom(T,X×SY)≅Hom(T,X)×Hom(T,S)Hom(T,Y).
On affine schemes the construction is concrete. Ring maps R→A and R→B give
SpecA×SpecRSpecB≅Spec(A⊗RB),
where A⊗RB is the tensor product. This affine calculation is why K⊗FK appears in the Galois torsor identity.