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Relations with "for each" composition and its properties (coming from profunctors with end c...

Intuitively, the composition $\sq$ is quite weird, although its categorical origin is at least natural: categorically, $\diamond$ is a 0-categorical analogue of the coend composition of profunctors, where … (This is slightly outside of the scope of the question, but I'm also interested in profunctors with end composition (and the properties $\mathsf{Prof}$ enjoys with such a composition). …
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