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I am trying to understand Street's proof Powerful Functors. In particular, proving that for a functor $p \colon E \to B$. The following are equivalent

  1. the functor $p^* \colon \mathbf{Cat}/B \to \mathbf{Cat}/E$ has a right adjoint.
  2. $p$ is a Conduche fibration

The beginning of the proof seems complicated. He uses coend calculus and later monadic functors. However, P. Johnstone uncomplicatedly (without coend calculus and monadic functors) showed that $(2) \implies (1)$.

I would like assistance with the intuition of coend calculus and monadic functors in relation to this. Perhaps a little bit of the explanation of the proof if possible.

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    $\begingroup$ Could you explain which part of the proof specifically you are having trouble understanding? I don't see any coend calculus, for instance, and Street gives a precise reference for the monadicity result he uses. $\endgroup$
    – varkor
    Commented Aug 10 at 12:34
  • $\begingroup$ Where he defines the category $\mathbf{Mod}$ with composition as being a coend formula. He uses this for the rest of the proof. $\endgroup$
    – Siya
    Commented Aug 12 at 10:00
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    $\begingroup$ Are you just asking for the definition of composition in Mod? You can find it on the nLab page for Prof, which is another name for that category. $\endgroup$
    – Naïm Favier
    Commented Aug 13 at 12:41

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