element chasing in an abelian category

Let \(\mathcal{A}\) be an abelian category and \(I: \mathcal{I} \rightarrow \mathcal{A}\) a (possible) small diagram Then one can element chase to verify, that \(I\) commutes

Consider the full subcategory \(\mathcal{A}_{\mathrm{sub}}\) induced by small diagram in an abelian category embeds in a small abelian category taking the discretized diagram of \(I\) Then by Mitchell's embedding theorem there exists an ring \(R\) and a fully faithful functor

\begin{align*} \mathcal{F}: \mathcal{A}_{\mathrm{sub}} \rightarrow \mathrm{RMod} \end{align*}

since category RMod is concrete, one can element chase ??

TODO, unsure https://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem