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Okay, but in this setup are the functors between such categories precisely the linear transformations? I don't have much intuition for whether this Hom makes sense.
Right, but it has a stronger property than that, namely being in Z_p and not just Z_q. What characterizes this element of Z_p? (I guess it's the trace of left multiplication by the Teichmuller representative of a on Z_q. Is that all one can say?)
