If a card is exiled face-down by Morph, can your opponent look at it?

When a card is turned face-down due to the Morph mechanic, it becomes a 2/2 creature with no name, subtypes, or abilities, and its true identity is hidden. Crucially, only the card's controller (and owner) is permitted to look at the face-down card at any time. This is explicitly stated in CR 707.7: 'A player who has a face-down spell or permanent may look at it at any time. Other players may not.'

This protection exists specifically to preserve the strategic mystery of Morph. Your opponent cannot demand to see the card, use a card effect to reveal it (unless that effect specifically says it 'looks at' or 'reveals' face-down cards), or otherwise gain information about it outside of special card effects that explicitly allow it.

When a face-down permanent leaves the battlefield for any zone — including exile — it must be revealed to all players (CR 707.9). So the secret is kept only while the card remains face-down on the battlefield; once it moves, everyone finds out what it was.

Example: You have a face-down Ainok Survivalist on the battlefield via Morph. Your opponent cannot look at it to check if it's a Morph creature. However, if a removal spell destroys it, you must flip it face-up and reveal it as it goes to the graveyard.