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:

Operation	How it's defined
multiplication or product $\mu :G_{S}\times _{S}G_{S}\to G_{S}$	First, note that $G_{S}\times _{S}G_{S}$ is a disjoint union of copies of $S$ indexed by $G\times G$. We consider the usual multiplication $G\times G\to G$ and define the morphism $\mu :G_{S}\times G_{S}\to G_{S}$ as exhibiting the same behavior on the corresponding copies of $S$. That this is a morphism follows from the constraints on $S$.
identity element $e:S\to G_{S}$	This is the map from $S$ to $G$ that sends it to the copy of $S$ indexed by the identity element of $G$.
inverse map $\iota :G_{S}\to G_{S}$	This sends the copy of $S$ indexed by $g\in G$ to the copy indexed by $g^{-1}$, and does so for each $g\in G$.