Actually, since the codomain of your functor P is actually Set, the argument I gave---which would be valid for any suitably complete codomain---can be rewritten to avoid the use of weighted limits entirely, by expressing those necessary in this case as hom-sets in a functor category. It then becomes the single calculation that End(P(-,F-)) = [C^op x C, Set](Hom_C, P.(1 x F)) = [C^op x D, Set](Hom_C.(1 x U), P) = [C^op x D, Set](Hom_D.(F^op x 1), P) = [C^op x D, Set](Hom_D, P.(U^op x 1)) = End(P(U-,-)) which requires no machinery beyond that of transpositions under adjunction. Richard On 7 February 2011 15:35, Richard Garner <richard.garner@mq.edu.au> wrote:

Dear Paul,

I do not know anywhere that it appears explicitly, but it can be pieced together quite quickly from results about weighted limits in Kelly's book. First, given any adjunction X -| Y : B --> A, any W : A --> Set, and any G : B --> C, we have

{WY, G} = {W, GX}  (**)

in the sense that the one exists if the other does, and the canonical comparison is then an isomorphism. This follows since Lan_X(W) = WY (by (4.28) of Kelly) and {Lan_X(W), G} = {W, GX} (by (4.58) ibid).

Since the end of a functor T: K^op x K --> E is by definition ((3.59) ibid) the limit of H weighted by the hom-functor H_K: K^op x K --> Set, we have, in the situation you describe, that

End(P(-,F-)) = {H_C, P.(1 x F)} = {H_C.(1 x U), P} = {H_D.(F^op x 1), P} = {H_D, P.(U^op x 1)} = End(P(U-,-))

by applying (**) twice to the adjointnesses 1 x F -| 1 x U  and  U^op x 1 -| F^op x 1, and using the natural isomorphism H_C.(1 x U) = H_D.(F^op x 1) obtained from the adjointness F -| U.

Richard

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