Is there a nice/unique way to express the pseudofunctor laws (http://ncatlab.org/nlab/show/pseudofunctor) when you don't have horizontal composition, only whiskering? I don't want to define horizontal composition because it's only unique up to homotopy, as per the argument in chapter 9 of the HoTT book.
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$\begingroup$ Horizontal composition not well defined? $\endgroup$– Fernando MuroSep 3, 2013 at 6:42
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$\begingroup$ It's not that it's not well-defined (it's definable in two ways, which are homotopic to each other), just that there's an arbitrary choice to be made, and I'm following a precedent in refusing to make that choice. See the note after lemma 9.2.8, which proves the interchange law for whiskering and natural transformation composition, at books.google.nl/…. $\endgroup$– Jason GrossSep 3, 2013 at 7:35
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$\begingroup$ Whiskering are horizontal composition of two cell (where one is a identity, identifiable by an arrow). From the note 92.8 I guess that may be you mean the GOdement law don't not worth (up to isomorphism or canonical morphisms). This is the case of categories enriched in the J.W.Gray monoidal structure, see LNM 391 $\endgroup$– Buschi SergioSep 3, 2013 at 14:20
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I did not realize that all the horizontal compositions listed on the nlab page had the identity on one side or the other. Since whiskering is equivalent to horizontal composition with the identity, it's easy to rephrase all the horizontal compositions as whiskerings.