Let me just mention the original question for this topic which is about the quotient of a scheme by a finite group scheme action. In SGA 3, the (general) definition is as following: Consider a diagram

$$ X_1 { \xrightarrow[]{d_0} \atop \xrightarrow[d_1]{} } X_0 \xrightarrow{ \ p \ } Y$$

We call $(Y,p)$ is a quotient if $p \circ d_0 = p \circ d_1$ and for any $q: X_0 \rightarrow Z$ such that $q \circ d_0 = q \circ d_1$, there exists a unique $r: Y \rightarrow Z$ such that $q = r \circ p$. The existence of the quotient $Y$ is equivalent to the representability of the functor $K: T \rightarrow K(T) $, i.e $K=\mathrm{Hom}(Y,-)$, here $K(T)$ is the kernel of

$$ \mathrm{Hom}(X_0, T) { \xrightarrow[]{T(d_0)} \atop \xrightarrow[T(d_1)]{} } \mathrm{Hom}(X_1, T) $$

In SGA 3, it's proved that the quotient exists in some case.

On the other hand, on wikipedia(group scheme), it's written that:

"For a subgroup scheme H of a group scheme G, the functor that takes an S-scheme T to G(T)/H(T) is in general not a sheaf, and even its sheafification is in general not representable as a scheme. However, if H is finite, flat, and closed in G, then the quotient is representable, and admits a canonical left G-action by translation. If the restriction of this action to H is trivial, then H is said to be normal, and the quotient scheme admits a natural group law. Representability holds in many other cases, such as when H is closed in G and both are affine.[1]"

It looks like that the two definitons of quotient are different. The first one considers morphisms to an object $T$ and the second definition considers morphisms from $T$. The first definition seems more natural to me for the quotient.

My question is : are these two definitions equivalent, under the following assumptions:

$X_0 = G$ is a group scheme, $X_1 = H \times G$ for a finite closed subgroup scheme $H$, $d_0 = m$ being the induced morphism from the multiplication and $d_1$ is the second projection.

ps: When I tried to figure out how to give a multiplication on the quotient $G/H$ (of course, one needs the condition "normal"), I have the first definition of quotient in mind, and can't see why "...and admits a canonical left G-action by translation". Using the second definition, it is easy to see it.