# Naturality up to (inner) automorphism?

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?

## 1 Answer

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.

• 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. Feb 15, 2020 at 19:53
• 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! Feb 15, 2020 at 20:33