Brouwer's theorem proof using Category theory

I am pretty new into Category theory and am currently reading the 2nd edition of "Conceptual Mathematics: A First Introduction to Categories" by F. William Lawvere and Stephen H. Schanuel and having some difficulties understanding the very start of the proof of Brouwer's theorem in the session 10 chapter 5.  1) Are the objects of this category really legit? From the all of the definitions of the category I understood that the objects are usually of the similar nature, i.e. finite sets, groups and etc., whereas here one object is physical arrows in the ball, other object is the ball itself and the third is the sphere...

2) When are you allowed to write a statement and say that this is an axiom? Here I am talking about Axiom 1 - it is just stated like that even though it is not obvious or a well known fact.

Thank you everybody for your help.

• I think $\mathcal{C}$ here is intended to be the category of topological spaces and continuous maps. The authors are a bit remiss in not explaining how the set $A$ of arrows is interpreted as a topological space. The "axiom" is a property distilled from Brouwer's proof that captures one of the important properties of the objects and maps used in the proof: it's a hypothesis about the map $p$ that happens to hold for the particular map $p$ used in the proof. Mar 20 '20 at 23:29
• Thank you for comments. Could you maybe also help me understand what is the T object? I know that there is a whole paragraph for it here but I dont get what is it meant with a "smooth listing" and what is the actual part of T in the proof. Mar 20 '20 at 23:53
• $T$ is some object or other with a map to $A$. "Smooth listing' is just referring to the fact that the map is continuous, I think. $T$ in this axiom is taken to be $S$ in the proof. Mar 21 '20 at 0:01