Constant group scheme

Definition
Suppose $$G$$ is a group and $$S$$ is a scheme which is the final object in a category of schemes that admits fiber products. The constant group scheme of $$G$$ over $$S$$, denoted $$G_S$$, can intuitively be thought of as a scheme that stores a copy of $$S$$ for each element of $$G$$, and operates on these copies as if they were the corresponding elements of $$G$$.

Explicitly, the scheme $$G_S$$ is a disjoint union of copies of $$S$$ indexed by elements of $$G$$, and the operations are as follows: