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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 11th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAn ordinary natural transformation F→G (F,G:C→D) can be converted into a functor I⨯C→D whose restrictions to {0}⨯C and {1}⨯D are F and G respectively. Here I={0→1}. It seems to me that a pseudonatural transformation can likewise be converted to a [[2-functor]] I⨯C→D. In particular, the morphism $\phi(f)$ in the definition of a [[pseudonatural transformation]] can be encoded via the compositor isomorphisms for the maps (0,r)→(0,s)→(1,s). Has this been written up somewhere? <a href="https://ncatlab.org/nlab/revision/diff/pseudonatural+transformation/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/pseudonatural+transformation/14">v14</a>, <a href="https://ncatlab.org/nlab/show/pseudonatural+transformation">current</a>

   An ordinary natural transformation F→G (F,G:C→D) can be converted into a functor I⨯C→D whose restrictions to {0}⨯C and {1}⨯D are F and G respectively. Here I={0→1}.

   It seems to me that a pseudonatural transformation can likewise be converted to a 2-functor I⨯C→D. In particular, the morphism $\phi(f)$ in the definition of a pseudonatural transformation can be encoded via the compositor isomorphisms for the maps (0,r)→(0,s)→(1,s).

   Has this been written up somewhere?

   diff, v14, current

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJun 12th 2022
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   Author: Mike Shulman
   Format: MarkdownItexIf you're saying "pseudonatural transformation" then you should probably say "pseudofunctor" to match. The small number of people who say "2-functor" to mean "weak 2-functor" probably also say "2-natural transformation" or just "natural transformation" to refer to the weak case. Anyway, this is certainly "well-known", but I don't know a reference offhand, sorry.

   If you’re saying “pseudonatural transformation” then you should probably say “pseudofunctor” to match. The small number of people who say “2-functor” to mean “weak 2-functor” probably also say “2-natural transformation” or just “natural transformation” to refer to the weak case.

   Anyway, this is certainly “well-known”, but I don’t know a reference offhand, sorry.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 12th 2022
   • (edited Jun 12th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexEssentially this is asking for (a reference for) the cartesian closure of $2Cat^{ps}$, for this implies that $$ \Big\{ \mathcal{C} \times \{0 \to 1\} \xrightarrow{\;\;\;\;} \mathcal{D} \Big\} \;\;\; \xleftrightarrow{\phantom{---}} \;\;\; \Big\{ \{0 \to 1\} \xrightarrow{\;\;\;\;} 2Cat^{ps} \big( \mathcal{C} ,\, \mathcal{D} \big) \Big\} \,. $$ All these "well-known" facts ought to be presented in any review of 2-category theory (and they ought to be stated on nLab pages), but (I looked through those we list at *[[2-category]]*) they may all be [[folklore]] instead . But people like to discuss something at least close, namely the *[[Gray tensor product]]*, which is a hack to make the above work while sticking to strict 2-categories. From the respective discussion in * [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378)) one can get close to (a reference for) the above pseudo-cartesian closure by combining their Thm. 12.2.20 with their Prop. 12.2.27 and then using the similarly "well known" fact that $\mathcal{C} \otimes_{\Gray} \mathcal{D} \,\simeq\, \mathcal{C} \times \mathcal{D}$ as bi-(i.e. weak-)2-categories. (I am not saying this is a satisfactory answer to the question, just indicating that this gets at least close. Probably one has to keep chasing through literature on that Gray-business to stitch together a reference.)

   Essentially this is asking for (a reference for) the cartesian closure of $2Cat^{ps}$, for this implies that

   $$\Big\{ \mathcal{C} \times \{0 \to 1\} \xrightarrow{\;\;\;\;} \mathcal{D} \Big\} \;\;\; \xleftrightarrow{\phantom{---}} \;\;\; \Big\{ \{0 \to 1\} \xrightarrow{\;\;\;\;} 2Cat^{ps} \big( \mathcal{C} ,\, \mathcal{D} \big) \Big\} \,.$$

   All these “well-known” facts ought to be presented in any review of 2-category theory (and they ought to be stated on nLab pages), but (I looked through those we list at 2-category) they may all be folklore instead .

   But people like to discuss something at least close, namely the Gray tensor product, which is a hack to make the above work while sticking to strict 2-categories.

   From the respective discussion in

   • Niles Johnson, Donald Yau, 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

   one can get close to (a reference for) the above pseudo-cartesian closure by combining their Thm. 12.2.20 with their Prop. 12.2.27 and then using the similarly “well known” fact that $\mathcal{C} \otimes_{\Gray} \mathcal{D} \,\simeq\, \mathcal{C} \times \mathcal{D}$ as bi-(i.e. weak-)2-categories.

   (I am not saying this is a satisfactory answer to the question, just indicating that this gets at least close. Probably one has to keep chasing through literature on that Gray-business to stitch together a reference.)