To be fair, Trump started it. Before Pope Francis arrived in Mexico last week, the U.S. Republican presidential candidate lectured him about problems with immigration and security at the U.S. border. “I think that the pope is a very political person,” Trump said. “I think he doesn’t understand the problems that our country has.”  

The Vatican could have chosen to swat this away, dismissing Trump’s importance as a political figure or ignoring his comments altogether. Instead, Francis and his communications team have savvily used the free publicity that seems to follow Trump to bring attention to the message of the pope’s trip: Christians and governments alike are called to care for migrants and the poor, including those who try to cross the Mexican border every day. On Wednesday, a Vatican spokesperson called Trump’s comments “very strange.” And on Thursday, Francis himself addressed Trump’s comments in a conversation with reporters on his return flight to Rome.

“A person who thinks only about building walls, wherever they may be, and not building bridges, is not Christian. This is not in the gospel,” he said, according to CNN.

Trump immediately hit back. “If and when the Vatican is attacked by ISIS, which as everyone knows is ISIS’s ultimate trophy,” he said in a statement on his website, “I can promise you that the Pope would have only wished and prayed that Donald Trump would have been President because this would not have happened.” He also suggested that the pope is a pawn of the Mexican government and said it’s “disgraceful” for “a religious leader to question a person’s faith.”

To go point by point: Yes, ISIS does often talk about taking down the “army of Rome”; as Graeme Wood reported last winter, “Who ‘Rome’ is, now that the pope has no army, remains a matter of debate.” No, the pope probably won’t be praying for a Trump presidency; when reporters on the Thursday flight asked whether U.S. Catholics should avoid voting for the Republican, the pope replied, “I am not going to get involved in that.” As for being a pawn of the Mexican government, it’s worth noting that the pope gently condemned the country’s “corruption, drug trade, exclusion of different cultures, violence, and also human trafficking, kidnapping, and death” during his opening address to government officials there.

As for the pope’s comments: You can make your own judgements about whether Trump is a Christian. But it is true, as Trump’s senior adviser Dan Scavino pointed out, that the Vatican itself is surrounded by walls.

The exchange, despite offering a riveting spectacle, seems unlikely to change much in U.S. electoral politics. The two men could not be more opposite in philosophy, disposition, or hairstyles—but they are leveraging each other’s celebrity to reinforce their own agendas. In South Carolina, where Trump is polling high and evangelical Protestants abound, the pope’s disparagement of the candidate’s faith seems more likely to help, not hurt, his chances.

And at least in the U.S. press, the pope’s zingers had what was probably an intended effect: They won immediate attention for his agenda in Mexico and a visit to the border that was covered somewhat demurely. Ever since Francis became pope in 2013, U.S. Catholics seem to have become accustomed to embracing the parts of him that they like and shrugging off the rest. If those voters were planning to support Trump before this kerfuffle, it seems unlikely that this exchange would compel them to switch their votes. For Catholics, as with other American religious groups, faith may or may not influence their pick of candidate: In a recent Pew poll, roughly similar percentages of Republican Catholics and Protestants thought Trump would make a good or great president.

The pope, being a man of peace, did make one concession to Trump. “As to whether I am a pawn, well, maybe, I don't know,” he said. “I'll leave that up to your judgment and that of the people.”