A permanent that turns face up or face down changes characteristics but is otherwise the same permanent. Spells and abilities that were targeting that permanent and Auras and Equipment that were attached to that permanent aren't affected unless the new characteristics of the object change the legality of those targets or attachments.
Because the permanent is on the battlefield both before and after it's turned face up, turning a permanent face up doesn't cause any enters-the-battlefield abilities to trigger.
If a face-down permanent leaves the battlefield, you must reveal it. You must also reveal all face-down spells and permanents you control if you leave the game or the game end.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Richard Whitters
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
Any time you have priority, you may turn a manifested creature face up by revealing that it's a creature card (ignoring any copy effects or type-changing effects that might be applying to it) and paying its mana cost. This is a special action. It doesn't use the stack and can't be responded to.
If something tries to turn a face-down instant or sorcery card on the battlefield face up, reveal that card to show all players it's an instant or sorcery card. The permanent remains on the battlefield face down. Abilities that trigger when a permanent turns face up won't trigger, because even though you revealed the card, it never turned face up.
To manifest a card, put it onto the battlefield face down. It becomes a 2/2 face-down creature card with no name, mana cost, or creature types. It's colorless and has a mana value of 0. Other effects that apply to the permanent can still grant it any characteristics it doesn't have or change the characteristics it does have.
Kozilek, the Broken Reality's triggered ability will resolve before Kozilek does. If Kozilek is countered or otherwise leaves the stack in response to that triggered ability, the triggered ability will still resolve as normal.
If a manifested creature would have morph or disguise if it were face up, you may also turn it face up by paying its morph cost or disguise cost.
You must ensure that your face-down spells and permanents can be easily differentiated from each other. You're not allowed to mix up the cards that represent them on the battlefield to confuse other players. The order in which they entered the battlefield should remain clear, as well as what ability caused them to be face down. (This includes morph, manifest, disguise, cloak, and a few other effects that turn cards face down.) Common methods for doing this include using markers or dice, or simply placing them in order on the battlefield.
If a double-faced card is manifested, it will be put onto the battlefield face down. While face down, it can't transform. If the front face of the card is a creature card, you can turn it face up by paying its mana cost. If you do, its front face will be up.
Turning a permanent face up or face down doesn't change whether that permanent is tapped or untapped.
If a targeted player has only one card in hand as Kozilek's triggered ability resolves, they manifest that card.
Because face-down creatures don't have a name, they can't have the same name as any other creature, even another face-down creature.
Unlike a face-down creature that was cast using a morph or disguise ability, a manifested creature may still be turned face up after it loses its abilities if it's a creature card.
