This category is Example 3.5 of Chapter 1 from Algebra, Chapter 0 by Aluffi.
Let $\mathcal C$ be a category and fix an object $A$ in the category.
Define a new category $\mathcal C_A$ by declaring the objects to be all morphisms from any object in $\mathcal C$ to $A$, and define the morphisms as follows:
Given two morphisms $f_1: Z_1 \to A$ and $f_2: Z_2 \to A$ in $\mathcal C_A$, a morphism between $f_1$ and $f_2$ in $\mathcal C_A$ is represented by a morphism $\sigma: Z_1 \to Z_2$ in $\mathcal C$ such that $f_1=f_2\circ \sigma$.
Is this really the only "sensible" way to define morphisms in this category? Are there other sensible ways to define morphisms in this category?
(I have a bit of a gripe since the author recommends the reader try to define the morphisms by themselves before reading on, and I couldn't figure it out.)