This question is about proposition 1.8.1 in Grothendieck's tohoku paper. (I am reading out of Michael Barr's translation).

The proposition says that if $\mathcal{A}$ is an abelian category which is cocomplete and and$$\sum (A_i \cap B) = B \cap \sum A_i$$ for all objects $A$, all subobjects $B$ of $A$ and lattices $\{A_i \}$ of subobjects of $A$ then filtered colimits are exact.

Grothendieck leaves the proof to the reader. Could anyone share some insight on how to prove the above claim?

Yesterday I asked this question on math stackexchange and Zhen Lin helped me sort out the converse, but I am still stuck with the proposition and no one seems interested in the question.

I hope that this question is suitable for mathoverflow.