Sorry, if my notation is weird. Let K denote the monoidal homotopy category of chain complexes over k. Given Kfunctors $$F: C \otimes D ^{op} \to K, G: D \otimes E^{op} \to K,$$ what is the explicit description of their composition as profunctors? I believe the abstract description in terms of tracing out with coend, http://ncatlab.org/nlab/show/profunctor (which I don't understand yet) in this particular case has something to do with the bar complex, and in appropriate sense Hochschild homology.
