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Consider a category $C$ and the corresponding metacategory $C'$ of slice categories $C/-$. That is, $C'$ has objects $\{C/x \mid x \in C\}$ and morphisms are functors $C/x \to C/y$. Let $F\colon C \to C'$ be a functor. In particular we map every morphism $m\colon x \to y$ in $C$ to a functor between slice categories $Fx$ and $Fy$. For every $m\colon x \to y$ in $C$, consider a natural transformation $\eta_m$ - from the "identity functor" $Fx \to C := Fx \overset{\mathrm{id}_{Fx}}{\to} Fx \overset{\pi_1}{\to} C$ - to the functor $Fx \overset{Fm}{\to} Fy \overset{\pi_2}{\to} C$ where $\pi_1,\pi_2$ are some canonical functors (that I leave unspecified, but in my concrete setting with one concrete $C$, they are pretty canonical). **Question:** The sentence above "For every $m$... consider a natural transformation" -- can we express it with more categorical language? E.g., is itself a functor or natural transformation somehow?
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Fri, 03 Sep 2021 15:39 GMT