Given a 2-groupoid $G$, two objects $a,b$, and two 1-morphisms $f,g:a\rightarrow b$ and a 2-morphism $\alpha : f \rightarrow g$, is it the case that
- there always exists a 2-morphism $\beta : f^{-1} \rightarrow g^{-1}$? Denote the operation that sends $\alpha$ to $\beta$ by $\texttt{flip}$.
- is it true that $\texttt{flip} \circ -^{-1} = -^{-1} \circ \texttt{flip}$ ?
- what is $\texttt{flip}$ usually called?
My guess is that this is true. However, I have scoured many sources (including many many pages over at the nLab), and I have not been able to find out if this is true. Perhaps this is only visible at level $3$, which is why this is not pointed out in anything about 2-groupoids?