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what does it mean for a function “not to respect equality”

I am reading from this book about different versions of the axiom of choice.

The author describes weak choice as the following

I have no clue what “respecting equality” means for a function. Surely, if two elements of the domain are actually equal, then they must map to the same element in the range by right uniqueness? Is the author abusing the definition of function to include relations which are not right unique? Or am I not understanding what is meant by ‘respect equality’?

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