Does the end determine the center of a coend?

Question

For distributors,if coends composes, does the end determine the center of such trace? More generally I guess, what is the end of a coend?

Answer 1

This request of clarification is too long to be a comment. :)

In case you're asking what is the end $$ \int_A\int^B P(A,B)\times Q(B,A) $$ for two composable profunctors $P,Q$ (necessarily in opposite directions)... no one knows in the general case. In some concrete cases, you can say something, for example if $P=Q=id$.

In general, this is a result of the kind "what is the limit of this colimit, and how does it relate to the colimit of the limit?". This kind of questions is notoriously difficult.