Transitioning from Cocoonababy (or the like) to crib mattress

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Hello world. We need advice on transitioning our LO from a reflux-helper mattress (Cocoonababy, nest,...) to a flat crib mattress.

LO is 11 weeks now and sleeps 6 to 8hrs comfortably in it - been doing that since 6weeks old. The Cocoonababy mattress should last another month or so but we want LO to start getting used to flat mattress already.

We tried going cold turkey one night and it didn’t go well. LO woke up every 1.5-2hrs. Currently trying to have day naps in cot.

How did you do it? How long did it take you?