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I'm curious about how Yoneda structures on a 2-category $\mathcal{K}$ play with various 2-categorical constructions. For example, if I have a 2-(co)monad on $\mathcal{K}$, are there any conditions guaranteeing the existence of a Yoneda structure on the 2-category of (co)algebras?

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    $\begingroup$ I have the impression that for a lex-comonad T on Cat, coAlg(T) has a Yoneda structure. The same for a lex-reflective subcategory of a 2-topos. $\endgroup$ May 19 '20 at 20:02
  • $\begingroup$ I can believe that, thanks! I actually found Charles Walker's paper and it had the sort of results I was looking for. $\endgroup$ May 19 '20 at 21:16
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    $\begingroup$ Proposition 4.4. in Weber's paper citeseerx.ist.psu.edu/viewdoc/… says that you can find a discrete opfibration classifier for each fully faithful subobject of a given one. The entire paper describes a procedure to generate a YS on $\cal K$ out of a dopfib classifier $\Omega$. $\endgroup$
    – fosco
    May 19 '20 at 21:28
  • $\begingroup$ Day convolution lifts the yoneda embedding for a monoidal category to a yoneda embedding in the 2-category of monoidal categories, and this formalises to liftings of yoneda embeddings of (co)lax algebras for well-behaved 2-monads T. Section 8 of link describes the case in which T extends to a double monad on an (augmented virtual) ``proarrow equipment''. For such T lifts of yoneda embeddings for colax T-algebras exist whenever T preserves composition of proarrows and both its unit and associativity cells satisfy certain Beck-Chevalley conditions. $\endgroup$ May 20 '20 at 20:46

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