I think it would be great if a moderator could delete both answers, as one is being heavily downvoted due to apparent trolling and the other only exists because the first lowered the user's reputation below the commenting threshold. Thanks.

Has the mystery been solved yet? What's up with these weird posts?

It's just some troll mimicking one of Angelo's answers. Is there something more mysterious about it?

Yes. The criterion for a symmetric monoidal functor I to be equivalent to the datum of such a fully dualized object is that it should have unramified diagonal (this is somewhere in Laumon Moret Bailly, I don't have it here). If I is from an n-cobordism to the n-category of n-families over a fixed symmetric n-category, the diagonal is a monomorphism, and a monomorphism is certainly unramified.

Has the mystery been solved yet? What's up with these weird posts?

whoever votes up answers that they don't understand?

I'm convinced that the answer is "lots of people".

Amen. And I speak as someone whose fairly banal observation about expressing the trace of a matrix as a weighted average of its numerical range continues to get occasional upvotes ;-)

Anyway, it really annoys me that somebody voted up that completely nonsensical answer, when I had a correct answer already posted (that still has no votes). Come on, guys (by which I mean the people who voted up that garbage), at least give credit where credit is due, and all that.

Edit: Something I in turn find curious is that now, that these examples have surfaced, nobody seems to be flagging them as spam. Anyway, I am going to.

Yes. The criterion for a vector bundle on a connected paracompact space to be invertible is that it should have unramified diagonal (this is somewhere in Laumon Moret Bailly, I don't have it here). If the vector bundle has a total Stiefel-Whitney class invertible in its cohomology ring, the diagonal is a monomorphism, and a monomorphism is certainly unramified.

In the comments, someone remarks that this is plagiarised. The original is here and the text is:

Yes. The criterion for an Artin stack to be Deligne-Mumford is that it should have unramified diagonal (this is somewhere in Laumon Moret Bailly, I don't have it here). If the stack is fibered in sets, the diagonal is a monomorphism, and a monomorphism is certainly unramified.

For comparison, there's the "answer" to Harry's question:

Yes. The criterion for f∧g to be in An is that it should have unramified diagonal (this is somewhere in Laumon Moret Bailly, I don't have it here). If An is a class of monomorphisms anodyne w.r.t. a separated segment on such a category of presheaves, the diagonal is a monomorphism, and a monomorphism is certainly unramified.

Two further occurrences of this remark:

1. http://mathoverflow.net/questions/42109/semicosimplicial-totalization
2. http://mathoverflow.net/questions/43537/property-t-and-subgroups-of-finite-index

(Apologies for cross-posting with Kevin - I got distracted by my hunt for further occurrences)

EDIT: I believe the answer Ben commented on was on this question, though it was deleted one way or another (could a 10k user or moderator confirm?)

(The original answer that provoked this has now been deleted.)

I've flagged the comment as spam, I urge others to do so.

(Note that the question itself is highly likely to be closed, but that is independent of troll-like behaviour)