# on composition of profunctors

Naive question, why is the equivalence relation necessary in the definition of composition of profunctors: http://en.wikipedia.org/wiki/Profunctor. Or if it is not necessary what is the advantage of adding the equivalence relation?

Edited: The remarks on bilinearity, and coherence answered my question. But let's suppose our ctegories are enriched over Vect, I am guess the composition would not be just the tensor product version of the definition, if so why not.

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Just because a Kan extention is a colimit then a quotient of a sum by equivalence relation. Otherwise the composition of simple functors isnt coherent by composition of the associate profunctors (a (simple) functors can view as a profunctor in covariant, or contravariant way). –  Buschi Sergio Dec 27 '11 at 8:26
Perhaps one can add that coherence is the norm and strict associativity is almost unnatural. –  Tim Porter Dec 27 '11 at 11:11