Circular definition in slice category?

Question

I am reading Aluffi (Algebra Chapter 0) there he introduces the slice category in a kind of excercise: When thinking about it I got confused about the "nature" of the $Z$ (and $A$). Since they are defined to be objects of the class Obj(${\sf C}_A)$ they should be morphisms as well. That seems kind of a circular definition. What am I missing?

Answer 1

I have missed that $A$ and $Z$ are not objects of ${\sf C}_A$ but only objects of ${\sf C}$, hence there is no circularity in the definition.