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I have fleshed out (and corrected) and then spelled out the proof of the statement (here) that Kan extension of an adjoint pair is an adjoint quadruple:
For $\mathcal{V}$ a symmetric closed monoidal category with all limits and colimits, let $\mathcal{C}$, $\mathcal{D}$ be two small $\mathcal{V}$-enriched categoriesand let
$\mathcal{C} \underoverset {\underset{p}{\longrightarrow}} {\overset{q}{\longleftarrow}} {\bot} \mathcal{D}$be a $\mathcal{V}$-enriched adjunction. Then there are $\mathcal{V}$-enriched natural isomorphisms
$(q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}]$ $(p^{op})^\ast \;\simeq\; Ran_{q^{op}} \;\colon\; [\mathcal{D}^{op},\mathcal{V}] \longrightarrow [\mathcal{C}^{op},\mathcal{V}]$between the precomposition on enriched presheaves with one functor and the left/right Kan extension of the other.
By essential uniqueness of adjoint functors, this means that the two Kan extension adjoint triples of $q$ and $p$
$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }$merge into an adjoint quadruple
$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]$Thanks! Fixed this also in geometry of physics – Categories and Toposes
Just for completeness, i have added (here) to the old coend-calculus proof of adjoint pairs Kan-extending to adjoint quadruples also a detailed proof using just colimit notation. (Either for pedagogical purposes, or because in this form it applies to $\infty$-category theory using only results available from standard sources).
For no good reason, I have typed out another elementary proof (here) that left Kan extensions of finite product preserving functors are themselves finite product preserving.
This simple proof does not mention pointwise sifted colimit-expressions for the Kan extension, but just uses the pullback-stability of colimits in the base topos and the Yoneda lemma over a large domain.
added also the non-coend-calculus proof that left Kan extension of fully faithful functors are again fully faithful (here).
(This stuff deserves to go to other entries, like Kan extension, but for the moment I keep them here.)
(In my local copy the TikZ diagrams are all spaced according to the golden ratio, but transporting them to here doesn’t preserve the spacings. I could fix that, but it’s not my priority now…)
Okay, I think I am done with doing some justice to the previously puny remark on cohesion, now an Examples-section here.
This contains now proposition and proof (here) that finite product preserving reflections of small categories induce cohesive adjoint quadruples on categories of presheaves, in a way that applies/works verbatim also in $\infty$-category theory using standard sources (i.e. no reference to enrichment and coends).
Among the examples this now mentions the cohesion of global- over G-equivariant homotopy theory.
I have extended statement and proofs (here and here) of Kan-extension-induced cohesive adjoint quadruples from adjoint pairs…
…in relaxing the assumption that the sites have finite products to the assumption that at least their free coproduct completion do so and that the coproduct-preserving extension of the left adjoint functor between them preserves these
(as that’s the generality needed for the cohesion of global- over G-equivariant homotopy theory for discrete $G$).
Of course the proof is even more direct in this case, since the assumption on the left adjoint is now “one step closer” to what needs to be proven. But the point is that this assumption is still readily checked in relevant examples.
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