I am trying to understand the semidirect product of groupoids, as defined in [this answer](https://mathoverflow.net/questions/23026/how-can-i-understand-the-groupoid-quotient-of-a-group-action-as-some-sort-of-p#23040) by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense, I am quite unfamiliar with that concept. So, as a test, I 'computed' the crossed product of two (arbitrary) sets $X,Y$ seen as discrete groupoids. My understanding is that the 2-group of automorphism in this case is the trivial monoidal category $(\mathbf{1}, \otimes, \ast)$, with $\mathbf{1}$ the trivial 1-object category, $\otimes:\mathbf{1}\times\mathbf{1}\rightarrow\mathbf{1}$, and unit the unique object $\ast$.


The object set of $X \rtimes Y$ is then $X \times Y$, and there is an arrow $(x\_1,y\_1) \rightarrow (x\_2,y\_2)$ iff $x\_1=y\_1$ and $x\_2=y\_2$, a long winded way of saying that in this particular case, $X \rtimes Y$ is (exactly) $X\times Y$ as a discrete groupoid. Question 1: is this correct?


There is another case that I have not been able to work as successfully: given an equivalence relation $E\_1$ on (set) $X$ seen as a groupoid, and equivalence relation $E\_2$ on $Y$, what are $E\_1 \rtimes E\_2$ and $E\_2 \rtimes E\_1$ ?


[Edit: I meant semidirect product, not crossed]



Semidirect product of groupoids
An action of a groupoid on another groupoid was defined in my paper "Groupoids as coefficients" Proc LMS (3) 25 (1972) 413-426, available [here](http://pages.bangor.ac.uk/~mas010/pdffiles/brown-gpdscoeff.pdf), which also uses methods of fibrations of groupoids. The term "coefficients" refers to nonabelian cohomology. This paper uses the term "split extension" instead of semidirect product. The more modern term is used given an action of a group on a groupoid in the book [Topology and Groupoids](http://pages.bangor.ac.uk/~mas010/topgpds.html), where it is relevant to the explicit description of *orbit groupoids*. Note that if a group acts on a space $X$, then it has an induced action on the fundamental groupoid $\pi\_1 X$, and also on the groupoid $\pi\_1(X,A)$ for any set $A$ of base points which is a union of orbits. 


