I'm not sure how to prove the definition Beta over the pre-category 1.8 (basically the exercise suggested by the book which is freely available so no problem: A gentle introduction to category theory by M Fokkinga) is a category. In particular I end up stuck even at the first point. When I try to verify point 1.3 I end up with
f and f' => A=A' and B=B'
Which of course does not seem to satisfy the axiom
In $\cal B$, a morphism is a triple: so a morphism $f$, when you apply 1.3 to $\cal B$ is actually $(A, g, B)$ for some morphism $g$ in the precategory $\cal A$ and then the third equivalence in the definition of $\cal B$ gives you what you want.
A typical example of a precategory has binary relations as the morphisms. Given the usual representation of binary relation as a set of pairs, you can't recover the intended domain and codomain of the relation. E.g., $R = \{(i, 0) : i \in \Bbb{Z}\}$ acts as a relation of type $X \to Y$ for any sets $X$ and $Y$ such that $\Bbb{Z} \subseteq X$ and $0 \in Y$. The construction in 1.8 gives you what we actually take for the category of sets and binary relations.The morphisms are triples like $(\Bbb{Z}, R, \Bbb{N})$ and $(\Bbb{R}, R, \Bbb{Z})$ so that you can read off the intended domain and codomain from the morphism.