dg.differential geometry – Reference request: isomorphic stacks are given by Moro Moro groupoids equivalent of Morita

Leave $ mathcal {G}, mathcal {H} $ The groupoids are lying. Leave $ B mathcal {G} $ denotes the main stack $ mathcal {G} $ packages and $ B mathcal {H} $ denotes the main stack $ mathcal {H} $ bunches Then, we have the following result.

The groupoids $ mathcal {G} $ Y $ mathcal {H} $ Morita is equivalent if and only if the differentiable batteries $ B mathcal {G} $ Y $ B mathcal {H} $ they are isomorphic

The only place where I could find proof of this is (theorem $ 2.26 $) the batteries and gerbes of Paper Differences of Kai Behrend and Ping Xu. I understand the test. I think it is not clear (the notion that something is clear depends on the level of experience) and needs some more lines in the test. I have written in detail the proof. It is a long test.

Is there any other place where this result has been tested so that I can quote that document? I have seen three articles that mention this result without any proof and all refer to the article by Kai Behrend and Ping Xu. I'm not asking for proofs. I'm just asking for reference.