Hi,  

In his book *(Categories for the working mathematician)* MacLane speaks (on page 45) about the category of objects (of $\textbf{Ab}$) under $\mathbb{Z}$ which is the comma category $(\mathbb{Z}\downarrow \textbf{Ab})$, and says *"it is the category of abelian groups with a selected element"* (in analogy with $(\star\downarrow\textbf{Set})$), but, is there not already a *prefered* (or *selected*) element in all (abelian) groups, namely the identity element?



The category of Abelian groups with selected elements
Pedro, sometimes we want to pick out an element or elements as an extra structure on abelian group. For example, we may be interested in a specified $\mathbb{Z}$-basis if the abelian group has one. Comma category constructions give a way of talking about such extra structure. 


Of course, as you say there is already a distinguished element given by the identity. This is reflected in the fact that there is a canonical functor 


$$Ab \simeq (\{0\} \downarrow Ab) \to (\mathbb{Z} \downarrow Ab)$$ 


obtained by postcomposing with $\mathbb{Z} \to \{0\}$. The comma category under $\mathbb{Z}$ gives a way of picking out *other* elements, if we want. 


