I am studying group schemes. I would say I understand the definition of an $S$-group scheme as to give an $S$-scheme together with morphisms for multiplication law, identity, and inverse and also the definition as representable functor from the category $(\text{Sch/S})$ to the category ($\text{Gr})$ of groups by Yoneda lemma. However, I do not know how to go from one to another in Examples. Here is one:

The additive group $\mathbb{G}_{a,R}$:

For a ring $R$, let $S = \text{Spec}(R)$, and $G= \mathbb{A}_k^1$.

For any $k$-scheme T, give $G(T) := h_G(T) = \Gamma(T,O_T) $ its additive group. This gives a group structure on $G$ with multiplication law

$$ m : h_G\times h_G \to h_G. $$

I want to write the multiplication law as a morphism of schemes $$ m_G: G\times_SG \to G. $$

I know there is construction involved Hopf algebra that gives the answer, but is there a way to obtain it using only Yoneda lemma? Below is my try. Is it right? is there a better way to do it? Is there something else to say?

Yoneda lemma says it should be the map $m(id_{G \times_S G})$. We should have $$ id_{G \times_S G} \in G\times _sG(G\times _s G) = R[X]\otimes R[X]. $$ Then from the fiber product universal property the scheme morphism $id_{G \times_S G}$ corresponds to the $R$-algebra homomorphism $$ (X \mapsto X \otimes 1) \otimes (X \mapsto 1 \otimes X). $$ Then the multiplication law is the scheme morphism that corresponds to the $R$-algebra homomorphism $$ \begin{align} \tilde{m} : R[X] & \to R[X] \otimes R[X], \\ X & \mapsto X \otimes 1 + 1 \otimes X. \end{align} $$


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