do the homomorphisms between two group schemes form a sheaf in the (whatever)-topology?

Question

By this I mean: Suppose you have two group schemes $G,H$ over a scheme $S$. Then you have a presheaf on the category $\text{Sch}/S$ sending $$(T\rightarrow S)\mapsto\text{Hom}_T(G_T,H_T)$$ which is a sheaf if $\text{Hom}$ means just normal morphisms of schemes (not homomorphisms of group schemes).

My question is: if Hom instead referred to homomorphisms of group schemes, then is the above still a sheaf?

I suppose this boils down to verifying the following:

1. Is the pullback of a group scheme homomorphism also a group scheme homomorphism?
2. If $T\rightarrow S$ is a covering in the (whatever)-topology, and $G_T\rightarrow H_T$ is a group scheme homomorphism whose restrictions to $G_{T\times_S T}$ agree, then is the gluing $G\rightarrow T$ also a group scheme homomorphism?

The first seems easy to verify, though I'm having trouble verifying the second...

Answer 1

As requested, here is the answer from my above comment:

1. This just follows from the fact that pull-back is functorial, and so the pull-backs commute with the pull-back of multiplication.

2. In any subcanonical topology $\mathcal{T}$- (e.g. anything coarser than the fpqc topology), you can check equality of morphisms locally. So, checking that your maps commute with multiplication is a $\mathcal{T}$-local condition.