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Let $C$ be a small poset category. I'm coming across a category where objects are (certain) functors valued in groups $F\colon C\to\operatorname{Grp}$, and morphisms between two objects $F$ and $G$ are almost natural transformations $\eta\colon F \Rightarrow G$. The difference is that if $f\colon e \to v$ is an arrow of $C$, we are allowed an inner automorphism $\Phi_e\colon Gv \to Gv$ such that the following diagram commutes $$\require{AMScd}\begin{CD} Fe @>\eta_e>> Ge \\ @VVFfV @VV\Phi_eGfV \\ Fv @>\eta_v>> Gv.\end{CD}$$

I'm curious if this "naturality up to automorphism" condition occurs elsewhere in the wild and if you know of a name for it?

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Assuming the inner automorphisms you get are assumed to satisfy some further coherency conditions, your morphisms should amount to pseudonatural transformations. This requires thinking of the category of groups as a 2-category.

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    $\begingroup$ To clarify: groups are seen as a $2$-category, where arrows are group morphisms, and two cells from $f$ to $h$ are element $g$ such that $g f g^{-1}= h$ (or the other way arround). It is equivalent to the $2$-category of connected groupoids. $\endgroup$ Feb 15, 2020 at 19:53
  • $\begingroup$ Oh this is very neat! I think the coherency conditions end up being vacuous in my setting ($C$ is a small category without loops coming from a graph), and I learned a new word! $\endgroup$ Feb 15, 2020 at 20:33

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