The method of straightening and unstraightening produces, for instance, an equivalence between left fibration and copresheaves. The construction seems very intricate. But what is the need for this construction? Are there functors that cannot be easily constructed directed as a copresheaf and we must resort to this straightening procedure? Or am I missing the point of this correspondence and it's supposed to be important for another reason?

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    $\begingroup$ A typical example is the construction of representable (co)presheaves. The simplicial formalism give you an easy access to slice and coslice fibration, but defining the corresponding (co)presheaves explicitly is much harder. Also I would say that the equivalence between fibration and presheaves is anyway an important result in category theory and you definitely want to be able to prove it in your theory of $\infty$-category... $\endgroup$ – Simon Henry Dec 13 '19 at 17:14
  • $\begingroup$ Hi I certainly don't doubt it's an important result. But can I clarify, you mean in these cases the fibration is easier to construct than the functor? Can you elaborate on why this functor to spaces is hard to construct? $\endgroup$ – davik Dec 13 '19 at 17:38
  • $\begingroup$ The slice or coslice fibrations are very easy to construct in the quasi-categorical framework (essentially, n-cells of the slice are just n+1-cells of the base whose 0 or n-th vertex is the object we slice on). I'm not saying it is impossible to give an explicite description of the corresponding functor, I'm just saying that I have not idea of how to do it, and that if you try to do it yourself you will probably struggle as well. $\endgroup$ – Simon Henry Dec 13 '19 at 18:07
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    $\begingroup$ The problem is that mapping spaces are not automatically functorial (essentially because you don't have a strict composition in quasicategories), so the usual formula for the represented presheaf doesn't define a functor. The construction of a "mapping space" functor and everything obtained from it, like the Yoneda embedding, is one of the main applications of straightening. $\endgroup$ – Achim Krause Dec 13 '19 at 21:03
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    $\begingroup$ To add to what Achim said, with our current technology (by which I mean HTT) there is no way to define the $\infty$ category of $\infty$ categories directly as a quasi-category (as far as I'm aware). Instead it is defined as a homotopy coherent nerve of a simplicial category (e.g. fibrant marked simplicial sets). So in HTT a functor into $Cat_{\infty}$ is encoded as a simplicially enriched functor. This forces us to leave the world of quasi-categories. To remedy this S/U allows us to talk about functors without leaving the world of quasi-categories by working with the unstraightned objects. $\endgroup$ – Saal Hardali Feb 3 '20 at 7:43

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