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I was wondering if there is a name for this 2-category which is like the 2-category of natural transformations, but does not actually require the 1-morphisms to be functors or the 2-morphisms to be natural transformations. That is

  • the objects are categories
  • the 1-morphisms between $\mathcal{C}$ and $\mathcal{D}$ are functions $Ob(\mathcal{C}) \to Ob(\mathcal{D})$
  • the 2-morphisms between $F, G : \mathcal{C} \to \mathcal{D}$ are assignments $\eta_c$, for each object $c$ of $\mathcal{C}$, to a morphism $Fc \to Gc$

For my application I am typically considering the case where the objects are (category product) powers of a single category, rather than all categories, but I suspect it does not make much difference.

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    $\begingroup$ Isn't this the category of natural transformation from $\mathcal{C}^d$ to $\mathcal{D}$ where $\mathcal{C}^d$ is the "discretization" of $\mathcal{C}$ (i.e. the category with the same objects but only the identity morphisms)? $\endgroup$ Feb 7 '14 at 18:37
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This is not a 2-category: there is no way to compose a 2-morphism $\eta : F\to G : C\to D$ with a function $H:Ob(D)\to Ob(E)$.

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  • $\begingroup$ Thanks, the two-dimensional structure of natural transformations is more sophisticated than I realised! $\endgroup$
    – Tom Ellis
    Feb 8 '14 at 10:37

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