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P.S. For the motivation of presheaf with transfers, I think (from reading MO) it's to accommodate (fully faithfully embed) the test space category with finite correspondences, which in turn is about ensuring say that Gysin maps in realization come from morphisms in the motive category; though I'm not
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I agree it should be a typo; X not Y. I have made a correction with a reference. —-
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