I'm just beginning to dabble into some category theory (from Aluffi's Algebra) and I have some difficulty with morphisms in a slice category. Particularly, I can't see why the diagram of a morphism should commute in the ambient category. If we have the category $C_A $ and two objects
$$ f:Z\longrightarrow A $$
$$ g:Y\longrightarrow A $$
why shouldn't any morphism $Z\longrightarrow Y$ count as valid, as long as there is a morphism from both $Y$ and $Z$ into $A$, not only those $\sigma$ such that $f=g\sigma$? If the answer is obvious or I have some fundamental misunderstanding you could just say so and I'll try to think more about it.