# 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?

• 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