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This is an elementary question on 2-categories, namely on the naturally arising notion of 2-adjunction when strict 2-functors are involved. Perhaps I don't understand some "internal" universal property of Kan extensions... (We restrict to the strict 2-category of categories and ignore size issues.)

  • One eventually encounters the universal property of the Yoneda embedding as a free cocompletion of a category. This seems to have three possible meanings (of interest). There's a forgetful strict 2-functor $U:\mathsf{CocompleteCat}\to\mathsf{Cat}$ from cocomplete categories and cocontinuous functors, and it may have some notion of left adjoint.

    1. There could be strictly 2-natural isomorphisms of categories $$\mathsf{CocompleteCat}(\widehat{\mathsf{C}},\mathsf D)\cong\mathsf{Cat}(\mathsf C,U\mathsf D).$$
    2. There could be strictly 2-natural equivalences of categories $$\mathsf{CocompleteCat}(\widehat{\mathsf{C}},\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$
    3. There could be 2-natural (pseudonatural) equivalences of categories $$\mathsf{CocompleteCat}(\widehat{\mathsf{C}},\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$
  • Given a category $\mathsf C$ one can consider its family fibration $\mathsf{Fam}(\mathsf C)\to \mathsf{Set}$. The objects of the domain are set-indexed families of objects of $\mathsf C$ while an arrow is given by a set-function between the indexing sets and a family of arrows in $\mathsf C$ in the obvious way. One may prove this assignment extends to a strict 2-functor $\mathsf{Fam}:\mathsf{Cat}\to \mathsf{Cat}$ which moreover lands in the category $\amalg$-$\mathsf{Cat}$ of categories with (small) coproducts and coproduct preserving functors. Again there's a forgetful strict 2-functor $U:\amalg$-$\mathsf{Cat}\to \mathsf{Cat}$ and again the $\mathsf{Fam}$ functor is the free coproduct cocompletion of $\mathsf C$. In fact it's even the extensive completion/envelope of $\mathsf C$. Thus once more three options arise.

    1. There could be strictly 2-natural isomorphisms of categories $$\mathsf{ExtCat}(\mathsf{Fam}(\mathsf{C}),\mathsf D)\cong\mathsf{Cat}(\mathsf C,U\mathsf D).$$
    2. There could be strictly 2-natural equivalences of categories $$\mathsf{ExtCat}(\mathsf{Fam}(\mathsf{C}),\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$
    3. There could be 2-natural (pseudonatural) equivalences of categories $$\mathsf{ExtCat}(\mathsf{Fam}(\mathsf{C}),\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$

What is the correct sense in which there are 2-adjunctions $\widehat{(-)}\dashv U$ and $\mathsf{Fam}\dashv U$? I think the correct answer should be the second option, since the non-strict notion is usually the "correct" but pseudonaturality of the equivalences should somehow disappear because the 2-categories and 2-functors are strict. Also I've never seen a formulation of the free cocompletion result with isomorphisms of categories. On the other hand both adjoints are given by Kan extension whose universal property gives bijection of sets, so I don't know how to get the equivalences..

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The bijection in the universal property of a Kan extension is a bijection between sets of natural transformations. Since all your options are fully faithful, this doesn't have any bearing on the question.

The restriction functor from cocontinuous maps out of a presheaf category to arbitrary maps out of the base is not an isomorphism, since the value of an extension on a colimit is determined up to isomorphism, not uniquely. It's possible to make the equivalence strictly 2-natural since we can make the Yoneda embedding a strictly natural transformation between the identity and the functor sending $J$ to its presheaf category. It is not possible, however, to make the inverse equivalence strictly 2-natural.

By the way, there is no adjunction here, in any sense, because of size issues.

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  • $\begingroup$ Thank you for your answer. I have two requests. 1. Could you explain why it isn't possible to make the inverse equivalence strictly 2 natural? 2. What about the $\mathsf{Fam}\dashv U$ 2-adjunction? $\endgroup$ – Arrow Nov 16 '17 at 12:50
  • $\begingroup$ I'm not sure whether it's actually a theorem that the inverse can't be made strict, but the reason there's no apparent way to do it is that it's built out of Kan extensions, rather than restrictions. Everything is the same for the families construction. $\endgroup$ – Kevin Carlson Nov 20 '17 at 16:01
  • $\begingroup$ I will try to understand the details. Thanks again. $\endgroup$ – Arrow Nov 20 '17 at 18:39

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