This question is a follow-up to my previous question Does the 'arrow' internal category forgetful functor have a right adjoint?, in which it is shown that the 'arrow' functor $Arr : Cat \to Set$ does not preserve coequalizers (and hence fails to have a right adjoint). However, it is true that $Arr : Cat \to Set$ preserves binary coproducts, because the set of arrows of a binary coproduct, i.e. disjoint union, of small categories is just the disjoint union of the two sets of arrows.
My question is whether this holds for a general category $\mathcal{E}$ with finite limits and binary coproducts: does $Arr : Cat(\mathcal{E}) \to \mathcal{E}$ always preserve binary coproducts? Or is there some such category $\mathcal{E}$ for which it does not?