Help with examples $3.9$ and $3.10$ in Aluffi's Algebra: Chapter $0$ I am studying Aluffi on my own, so I was hoping someone could check my thinking for examples $3.9$ and $3.10$ (p. $24$-$26$).
Example $\mathbf{3.9}$
Let $C$ be a category, $A, B$ be objects in $C$ and $\circ_{C}$ be composition in the category. We define $C_{A,B}$ to be a category as below:
Let $\text{Obj}(C_{A,B})=$ all $(f,g)\in \text{Hom}_{C}(Z,A)\times\text{Hom}_{C}(Z,B)$ for every object $Z$. If $O_{1}=(f_{1},g_{1})$ and $O_{2}=(f_{2},g_{2})$ are objects in $C_{A,B}$, then we define the set of morphisms between $O_{1}$ and $O_{2}$ as
$$\text{Hom}_{C_{A,B}}(O_{1},O_{2})=\{\sigma\in\text{Hom}_{C}(Z_{1},Z_{2})\mid f_{1}=f_{2}\circ_{C}\sigma\text{ and } g_{1}=g_{2}\circ_{C}\sigma\}.$$
Where $Z_{1}$ and $Z_{2}$ are the sources of $f_{1},g_{1}$ and $f_{2},g_{2}$, respectively. Composition in $C_{A,B}$ is inherited from $C$. That is, if $\sigma\in\text{Hom}(O_{1},O_{2})$ and $\tau\in\text{Hom}(O_{2},O_{3})$, then $\tau \sigma=\tau \circ_{C} \sigma$. By definition, $f_{1}=f_{2}\sigma$ and $f_{2}=f_{3}\tau$, which means that $f_{1}=f_{3}\tau\sigma$. Similarly, $g_{1}=g_{3}\tau\sigma$, thus $\tau\sigma\in \text{Hom}(O_{1},O_{3}).$ Associativity follows directly from the associativity of $\circ_{C}$.
If $O=(f,g)$ is an object in $C_{A,B}$ and both $f,g$ have source $Z$, then $1_{O}=1_{Z}$ (the identity morphisms are the identity morphisms in $C$). I believe I have constructed a category the way Aluffi was alluding to, but I just wanted to make sure.
Example $\mathbf{3.10}$
This just seems to be a carbon copy of example $3.9$, which is what I am concerned about. In fact, the only thing I would change is to include that $A$ is $\alpha$'s source and $B$ is $\beta$'s. I am not seeing any reason why the inclusion of $\alpha$ and $\beta$ changes anything.
Edit: I just realized that in order for $(f,g)$ to be an object, it would also need the additional restriction that $\alpha f=\beta g$.
I would also like to ask, do these categories have special names? Thank you!
 A: Your observations are correct. Example $3.10$ is a variation on example $3.9$ inducing more conditions on the given morphisms. There this simply means we add an object $C$ an require compatibility of the morphisms $f,g$ with the new morphisms $\alpha,\beta$ (compatibility=some diagrams commute).
Well, they have special names in some sense as they may be realized as instances of more general category-theoretic constructions (such as slice or comma categories). I do not think it will greatly help to elaborate on these, although I can if you want me to.
Think about these categories like this: They are templates for universal constructions (which is covered in §I.$5$ of Aluffi). By this I mean that we can use these categories to characterize certain objects.
For example, take sets $A,B$ and consider their cartesian product $A\times B$. If you take $Z=A\times B$ with $(f,g)=(\pi_A,\pi_B)$ the coordinate projections, then you can construct for any pair $(f,g)$ from a given object $Z$ a (unique) morphism in $C_{A,B}$ from $Z$ to $A\times B$: $Z\to A\times B,\,z\mapsto(f(z),g(z))$. We say that $(A\times B,\pi_A,\pi_B)$ is a terminal object of $C_{A,B}$. The analogue construction for the category $C_{\alpha,\beta}$ is called the fiber product.
Aluffi uses the categories $C_{A,B}$ and $C_{\alpha,\beta}$ as ad hoc constructions to introduce these concepts formally by means of diagrams valid in any category. This is in the end one of the main points of a category-theoretic-minded approach towards algebra.
