There are no political parties as such within the Vatican, so you may well be wondering why it is included in this series. It is true that, for obvious reasons, there are no PBP and CL factions within the Catholic Church. There is, however, an very important rift between groups that are recognizably CD and Reactionary that is pervasive within the Church today.
Pope Francis and Bono are CDs. In a nutshell, they believe in reaching out to the outside world and in love and compassion over the enforcement of rules. If the rules have to be fudged a bit in order to reach millions of people who desperately need help and guidance, so be it. Benedict and Ross Douthat are Reactionaries. They believe in drawing lines and enforcing traditional rules; if the rest of the world doesn’t like it, it’s their loss.
The difference between the two groups is perhaps most obvious in their attitudes towards immigrants. Francis always makes a point of showing favor to them; Douthat essentially agrees with Trump on this issue, as well as on abortion. That is the reason some passionate Catholics who otherwise wouldn’t give Trump the time of day, given his personal morality, voted for him in 2016.