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..

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.
• 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? – Arrow Nov 16 '17 at 12:50