The Church and the Clock Tower had their own little shadow war going on. But those of the Church and the Magi worked together sometimes to try and create a miracle.

Rin might be able to tell you a bit about it.

[Well, Caren WOULD make everyones life a misery, what pattern is that...]

Especially the breeding. She's the product of that. As for the product of Kirei Kotomine's marriage, no. He had a daughter. The daughter did not inherit his circuits, and gained something from her mother instead.