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. 



on composition of profunctors
Profunctor composition is not strictly associative even after putting on the equivalence relation. One is in a bicatgeory setting here. The equivalence relation is among other things to make identity profunctors work. You can think of profunctors as bimodules and composition as tensor product. The equivalence relation is the usual one for tensor products of bimodules. Associativity upto iso is because tensor product is associative up to isomorphism. 


