my question: does the morphism above imply the other way around, ie, y->x?
Well...if it's bijective...yes...and it's a homomorphism...I guess...[?]
there is a set of morphisms between any objects in your category. so while it is not correct to say that x-->y implies y-->x, it is true that there exists a set of morphisms (which might be trivial) from y-->x. but this is not dependent on the morphisms from x-->y
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