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The ability of Master's Manufactory can be activated no matter what happened to the artifact after it entered under your control. It may have left the battlefield, stopped being an artifact, or another player may have gained control of it.

If a card that isn't a transforming double-faced card becomes a copy of a card with craft, it'll stay in exile if you activate the craft ability. It won't return to the battlefield.

You may exile tokens you control as part of the materials required. However, because they aren't cards and won't stay in exile, any abilities that refer to what you "used to craft" the back faces won't refer to anything.

The back faces of some cards with craft refer to cards "used to craft" it. This refers to the cards exiled as part of the cost of the craft ability of the front face. Those cards are considered to be "used to craft" that permanent as long as they remain exiled and the permanent remains on the battlefield, even if the permanent's controller changes or some of its characteristics change (because of a copy effect, for example.)

If the materials required include multiple objects, you may exile some of them from among permanents you control and the rest from among cards in your graveyard. You don't have to choose all permanents or all cards from your graveyard.