Continuous map over some third space (trying to understand the stack)?

Question

I am trying to understand the notion of stack and that is why I am reading http://homepage.sns.it/vistoli/descent.pdf whose p.70 (first paragraph of 4.1.1) contains the example about 2 mappings: $f:X \to U$ and $g:Y \to U$ and then it mentions $\phi : X \to Y$ and calls it continous map over U. How to understand the phrase map over U? Is U used as some kind of additional argument? Maybe $\phi : X \to Y$ is the family of maps parameterized by this additional argument $\phi_i : X \to Y, i \in U$?

Update: Most likely it means $\phi : X \to Y \equiv f^{-1}(U) \to g^{-1}(U) $?

Answer 1

It means that $g\circ phi=f$. The notion of an overcategory is used implicitly throughout the text, but what the morphisms in the overcategory happen to be described in 3.6.1., page 60 in the pdf you're reading.