In [this paper](http://arxiv.org/pdf/math/0511710v2.pdf), the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a 2-space by considering only identity morphisms) by:
$$t\_j \bar{t\_i}(x,f)=(x,fg\_{ij}(x))$$
Where $(x,f)$ is a point in $U\_{ij}\times F$ for a double overlap $U\_{ij}$ of a cover $\{U\_i\}\_i$ of the base space $B$ and $F$ a 2-space (the fiber 2-space), $g\_{ij}(x)$ is an autoequivalence (mainly a functor) on $F$ and it "acts" on the right (as the custom wishes), $t\_i$ is an equivalence $t\_i:E|\_{U\_i}\rightarrow U\_i \times F$ ($E$ is the total space), and $\bar{t\_j}$ is the weak inverse of $t\_j$.


I found this definition rather obscure because $g\_{ij}(x)$ do have two maps (object and morphism ones), then there must be 2 different "transition functions" whith acts both on objects and morphisms, this definition suggests that $f$ is an object of $F$ and then, $g\_{ij}(x)$ is only the object part of the equivalence, I don't think such authors forgot the morphism map in their article, which leads me to think that I missed something, but what?



A question on 2-bundles
They don't say that $f$ is an object of $F$, so it might as well also be a morphism of $F$. In other words, the same equation defines both the object and the morphism parts of $g\_{i j}(x)$, with $f$ interpreted as either an object or a morphism respectively.


