Once you enter the penitentiary and use the boost rings for the first time, you will come to a blue door on the right up high. Use the Aquila to cut through and then grapple over to reach the key.
When you reach the divinity statue, grapple over to the catwalk on the right. Follow it to the right side and you will come across the key inside the building.
After you smash through the second red jewel in the elevator, grapple upwards in the next room to the doorway. Turn around when in the doorway and you’ll see the key across from you.
After you use the boost rings for the first time and go to the left where a large tv screen blocks your path, there will be a fenced are across from you. Pull out the container and jump on it. Then jump to the right of the fenced area to find an opening you can go through to reach the door there.
After you hit the second red jewel in the elevator, this door will be to the left of you when you land and fight the Rages.
After you have defeated the enemies where the train I running long the ceiling, go to the opening at the far end and make a right when you can. There you can find a blue door to cut through. Once you are through it, the key will be at the end of the corridor.
Just after the start of the mission, you will hop into a tunnel that extends. Look to the upper right and you will see a blue door that you can use the Aquila to cut open. Hop up to the left ledge and use the Aquila and then hop over to the opening and go to the right to find the door.
As soon as you find the first divinity statue, continue down the tunnel and on the left there will be an opening with a red door. Hop up into it and smash it with the Arbiter to find the door.
Just after you have received the devil trigger, you will enter a building and battle against some Rages. To get the key, you will need to launch an enemy into the air and then use your grapple to launch off them and jump up to the balcony above. Once you get to the balcony, you can find the key in the corner.
Just after you receive the Devil Trigger, you will enter a building where you will be introduced to the Fallen Harpys. Once you have defeated them and the Witches that appear as well, the door will become accessible in the far corner.