Consider a 2-group (seen as a 2-category with only one object $\star$) constructed from a crossed module $(G,H,t,\rhd)$ ($\circ$ will denote 1-morphisms composition, $\circ_{h}$ 2-morphisms horizontal composition, $\circ_{v}$ 2-morphisms vertical composition, when composed using them, symboles will mean (1,2)-morphisms, otherwise, they will simply mean elements of the groups $G$ and $H$), choosing a given convention we get horizontal composition of 1-morphisms:

with $gg'=g\circ g'$ is done using composition law of $G$. We get also, using the same convention, horizontal compostion of 2-morphisms:

with $\tilde{h}=h_{1}(g_{1}\rhd h_{2})=h_{1}\circ_{h}h_{2}$ is done using both action of $G$ on $H$ and compostion law in $H$. In the two precedent compositions, “order” of elements is preserved when composing. Why then the same thing does not happen in vertical composition of 2-morphisms? Indeed,

where $h'h=h\circ_{v}h'$ is done with composition law in $H$. Knowing that, $\circ_{v}$ has the same “origin” as $\circ$, in the sens that, $\circ_{v}$ is the composition law (function) in the category internal to the category having $\circ$ as its composition law (functor).