A question on 2-bundles

In this paper, 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?

• So another line that has $t_j \bar{t}_i ( ( x,f) \to (x , f') ) = ( x , (f \to f') g_{ij} (x) )$? – AHusain Dec 26 '15 at 22:13
• What does this line say? – Pedro Dec 27 '15 at 8:55

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.