When we first meet Pope Adrian in Act 3, Scene 1, he doesn't exactly make a great first impression. He's too busy commanding rival Saxon pope Bruno to get down on all fours so he can use him as a stepstool.
Is this Marlowe looking for laughs? For sure. But arrogance and pride are also par for the course with this character, who, most scholars agree, is meant to symbolize the Catholic Church from the point of view of Protestant England.
Throughout his scene, the Pope displays what Protestants of this time period would have thought of as typically "Catholic" behaviors. He uses a bunch of man-made books (rather than just the Bible) to pass judgment on Bruno, makes threats of excommunication, brags about his power to condemn or save his fellow Christians, displays a belief in the existence of Purgatory (a state of salvation somewhere in between heaven and hell), crosses himself frequently, and, to cap it all off, has his friars perform an exorcism of the spirit he thinks is haunting him—Faustus.
To a Protestant of Marlowe's day, Pope Adrian would be Catholic with a capital C. Most of his behaviors appear totally ridiculous, which makes Catholicism in general appear ridiculous, too.
It doesn't help that the guy is totally illogical, either. When Bruno points out that a prior Pope recognized the sovereignty of the Holy Roman Emperor, Pope Adrian just says something along the lines of, "Well, he was wrong. That prior Pope must have decreed in error. And I decree that he was wrong because I am the Pope, and I am invoking the doctrine of papal infallibility, which means that a Pope's decrees are never wrong." Except, you know, he says it in a prettier way, because Christopher Marlowe was putting the words in his mouth.
But here's the thing. Pope Adrian has proven this prior Pope wrong by invoking papal infallibility, which says that Popes are never wrong. That's not the most logically sound of arguments, now is it?
In Act 3, Scene 1, the Pope appears ridiculous, logically inconsistent, arrogant, and proud. This characterization means that the audience will delight in seeing him tormented as Faustus foils his plans to punish Bruno then steals all his good silver and china.
But it also makes us wonder why Faustus even bothers to torment him at all. The guy doesn't seem like a very formidable enemy, after all. This character and his storyline may just be for comedic effect, but they also give us important information about the priorities of Faustus's character and Renaissance stereotypes of Catholicism. In short, Faustus is petty, and Protestant Brits were not fond of continental Catholics (to generalize wildly).
Bruno is a schismatic Pope. In the wayback days, the Catholic church experienced all kinds of schisms, or divisions when different members disagreed on certain issues. The disagreeing portions of the church would often appoint their own popes--like Bruno.
Of course we don't know much about Bruno, and he's not much of a character in his own right. But his impact on Faustus is huge. Mainly to annoy Pope Adrian, Faustus decides to free Bruno from his punishment at the hands of the ridiculous Pope. But when Faustus returns to Germany with Bruno, having saved him from what was sure to be torture and other awful things, the Emperor is incredibly grateful.
Since we know that Pope Adrian is a bit of a joke, we're inclined to side with Bruno and his Emperor when it comes to whatever they're squabbling with Pope Adrian about. So we're kind of glad that Faustus rescues this guy, even though his motivations are less than good. Still, it's important to realize that Faustus manages to accomplish something good, or at the very least commendable, with his powers. We're betting he didn't mean for that to happen.
Plus, this whole scene with Popes Adrian and Bruno pokes some fun at what many Protestants saw as the illogic and corruption of the Catholic Church in the wayback days. Bruno seems like a good guy when compared to Pope Adrian. But we don't actually know if he's a good guy. We just know he's not Adrian, and that appears to be good enough for Faustus.