Why was this cyborg sent back and who sent it?
The Terminator Wiki makes the claim that Connor might have sent it. If Skynet sent it why did they care about John Henry?
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There's some discussion on this topic on the Episode 219 talk page and the Catherine Weaver talk page. The generally accepted idea is that there are likely three factions: "classic" Skynet (out to kill the resistance and go about it's usual Skynet business), "new" Skynet (John Henry) and some/one T-1001s (liquid metal Terminators), or the Resistance.
It seems that Catherine Weaver was likely sent back to ensure the creation and survival of John Henry. As to who sent it, it would seem advantageous to either the Resistance (maybe John Connor specifically, maybe not, depending on the timeline) or to "new" Skynet. If it was "new" Skynet, though, it implies some type of time-loop causality - John Henry basically sent Weaver back to make sure he gets created (though that's not particularly far-fetched for Terminator).
Katherine Weaver's mission is to make sure that John Henry survives.
There is clearly a second AI that sprung up around the same time as Skynet. It doesn't exactly hold human life to be sacred (as seen by the number of people Catherine Weaver kills), but it also isn't totally hell bent on the destruction of the human race. It also is at the same technological level as Skynet, as it is able to make (or use) T-1001s (or it suggests that John Henry takes over Terminators (much like "he" used T-888 body and Cameron's chip)).
John Connor sends an emmisary to this different AI asking if it will join their fight.
"Tell John Connor: the answer is no."
Why send "her" back? There is clearly a lot more time-traveling and time-manipulation happening than is suggested in the movies. It seems like every third person you meet is either a resistance fighter or a terminator (making only 1/3 of people in the past indigenous humans). Future John Henry probably saw a danger where "he" might not exist or be at a severe disadvantage and wanted to ensure the best future for himself.