Does the morphism above imply the other way around, ie, y->x?

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The discussion centers on whether a morphism y->x can be inferred from x->y. It is clarified that while a bijective morphism allows for this implication, it is not universally applicable. The existence of morphisms between objects in a category does not guarantee a direct reverse relationship. Instead, there may be a set of morphisms from y to x, which could be trivial. The relationship between morphisms is nuanced and not simply reciprocal.
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my question: does the morphism above imply the other way around, ie, y->x?
 
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Well...if it's bijective...yes...and it's a homomorphism...I guess...[?]
 
Originally posted by loop quantum gravity
my question: does the morphism above imply the other way around, ie, y->x?

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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