This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-Popescu. I was sure I had seen $(1)$ in Bourceux's encyclopedia, but I can't find it anymore.

- The category of internal abelian group objects $\mathsf{Ab}(\mathcal{E})$ is a Grothendieck category.
- Call $\mathsf{Set}[\mathsf{Ab}]$ the classifying topos of abelian groups, and let $\mathcal{E} \simeq \mathsf{Sh}(C,J)$. Then $$\mathsf{Ab}(\mathcal{E}) \simeq \mathsf{Cocontlex(\mathsf{Set}[\mathsf{Ab}], \mathcal{E})} \simeq \mathsf{lex}(\mathsf{Ab}_\omega,\mathcal{E}) \simeq \mathsf{lex}(\mathsf{Ab}_\omega,\mathsf{Sh}(C,J)) \simeq \mathsf{Sh}(C,\mathsf{Ab}).$$