Continuity mistake: After the baby is born, the nurse puts a blue towel on her, completely covering her chest. The next shot, the baby's chest isn't covered. (00:06:00)
Continuity mistake: At the beginning of the movie, when Arnold and the Stryker Security crew are protecting the man in the limo, pay attention to the crowd. Right when they arrive at the building, and begin to get out the vehicles the pedestrians begin to duck and run for no apparent reason. Especially noticeable in the shot, right before Arnold says "We're exposed". (00:16:30)
Continuity mistake: When Satan gets out of the car and turns up his collar, he has no gloves on his hands, but in the next shot as he is still turning up his collar, his hands are already gloved. (00:16:55)
Continuity mistake: Just when Jericho catches the priest jumping off the rooftop with Jericho hanging from the helicopter. in the next shot, his partner yells, "He's Got Him!" When the shot goes back to Jericho, he's hanging from the line with no priest in his hands. The next shot shows him hanging from the helicopter, holding on to the priest. (00:20:25)
Continuity mistake: In the subway scene, after Christine has given the bum the money and he shatters, his face hits the floor with the top of his head towards Christine, and his face away from her, when the shot changes to a close-up, his head has turned, and is facing her. (00:30:25)
Continuity mistake: After the priest writes down 666 on a piece of paper, he holds it upside down to reveal that it is 999. When we cut back to him a few seconds later, the numbers look completely different. (01:05:40)
Continuity mistake: When Satan shows Arnold the bandits who killed his family, Arnold tries to catch one of them, falls through the bandit, toppling the Christmas tree. But when Satan changes the flat back to normal (with no tree) we can see the tree standing upright as it disappears. (01:12:20)
Continuity mistake: When Satan enters the church demanding to know where Christine York is, one of the priests approaches him with a crucifix. The camera is facing the priest, Satan grabs the crucifix and stabs the priest right in the centre of his forehead (just above the eyebrow line). The camera angle changes to facing Satan and when the priest hits the deck, the crucifix is sticking out of his head higher up and above his left eyebrow. (01:22:30)
Continuity mistake: In the scene were Arnold disconnects the carriages of the train, near the end, he has to jump across the gap. When we first see the gap it is a certain distance, the camera angle changes and it's a bigger gap, then the angle changes and it's closer, there is no way Arnold could have jumped that big gap. (01:42:35)
Continuity mistake: In the train sequence near the end as Arnold shoots Satan out of the train, he flies toward the other carriage and starts to fall short, then in the next shot he lands ass first in the train. (01:43:10)
Continuity mistake: In the church at the end when the windows are shattering, Arnie's got his right arm protecting his face in a wide shot and a closeup, then in the next shot (behind him) his left arm's up. we then cut in front of him again and his right arm's back up. (01:44:20)
Continuity mistake: When Arnie is in the church the things on the tables keep changing - in one shot we candles on it but in the next we see a cross. (01:46:25)
Continuity mistake: Arnold fires a grenade at the devil, which knocks the devil back into his own train-cart, sliding him all the way back to the end. However, it's the front of the cart that explodes.
Answer: The Vatican is aware of the birth, somewhere in the world, of Christine (though not knowing her identity at the time), and the Satanists are present at her birth and know that she is "the one." It's safe to assume that the Vatican, and the Catholic Church in general, has spies/informants in the Satanists' society, and were able, in the intervening 20 years between Christine's birth and the events of the film, to track her down to New York and send Aquinas there (the fact that he was sent by the Vatican is established).