Consider the definition of group scheme in *Stack Project* [[022R]](https://stacks.math.columbia.edu/tag/022R). In the paragraph following definition 39.4.1, it is said that



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> We have morphisms of schemes over $S$: (identity) $e:S\rightarrow G$ and (inverse) $i:G\rightarrow G$ such that for every $T$ the quadruple $(G(T)=Hom(T,G),m,e,i)$ satisfies the axioms of a group listed above.
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As for my intuition, the identity is the identity of $G(S)$, and the inverse is the inverse of $id : G\rightarrow G$ in $G(G)$. But I don't know how to check that $(G(T),m,e,i)$ is a group.



Definition of group scheme
You need to use the condition that $m$ is a morphism of scheme. By that we can deduce $G(\cdot)$ is a functor.


