TITLE: How to get used to commutative diagrams? (the case of products).
QUESTION [7 upvotes]: I've returned to Aluffi's book (after getting the basics of groups from Herstein's) and I hit the same brick wall that made me put it aside. My general problem is this: 
I can't seem to get used to commutative diagrams. This is mainly due to the fact that i'm not sure I'm translating them correctly and there's no one to tell me when I do. I'd like to know what should I do to become more comfortable with reading and writing diagrams. 
And now for a specific example of me trying to translate a diagram about  the definition of products:

This is my parsing of the definition of products:
Let $\mathrm{C}$ be a category and Let $A, B \in Obj(\mathrm{C})$.
Let $\mathrm{C'}$ be a category obtained from $\mathrm{C}$ as follows:
Take the objects of $C'$ to be tuples $f = (f_A,f_B)$ where $f_A \in Hom_{\mathrm{C}}(F,A)$ for some $F \in Obj(\mathrm{C})$ (and $f_B \in Hom(F,B)$)
For every $f,g \in \mathrm{C'}$ let $\sigma \in Hom_{\mathrm{C'}}(f,g)$ iff $\sigma  \in Hom_{\mathrm{C}}(F,G)$ (where $F$ and $G$ are the "domains" of $f$ and $g$ respectively) and $f \circ \sigma = g$
 (I can't seem to formulate the definition without requiring that the morphisms themselves have some kind of product. what is happening here exactly? 2-categories?).
If $\mathrm{C'}$ has a final object $\pi = (\pi_A,\pi_B)$ it is said that $\mathrm{C}$ has a product object $A \times B$ which is the "domain" of $\pi$. 
Is this a right way for interpreting the univeral property of the product? 
ADDED: By "domain" of $f \in Hom_{\mathrm{C'}}(A,B)$ I mean the object $F$ in $\mathrm{C}$ such that $f_A \in Hom_{\mathrm{C}}(F,A)$ and $f_B \in Hom_{\mathrm{C}}(F,B)$.

REPLY [4 votes]: I don't have Alufi's book to verify, but from what I see in this topic, he defines the product as a universal cone. But it specifies the notions to the point that it is unhelpful.

So let's go to full generality and then apply to the case of the cartesian product. (I do not say it is the easiest way to get cartesian products, but this is the general construction over diagrams that Alufi is miming with its triples and stuff.)
Let $J$ be a small category, and $C$ any category. A commutative diagram indexed by $J$ in $C$ is a (covariant) functor $J \to C$. For every object $c$ of $C$, there is a constant diagram indexed by $J$ in $C$ defined by
$$ j \mapsto c, \quad (j\to k) \mapsto \mathrm{id}_c .$$
We denote  it by $\Delta_{J,c}$.
Then fix a diagram $D \colon J \to C$. From the category of functors $[J,C]$, we get the slice category $[J,C]/D$. Let denote $\mathrm{Cone}(D)$ the full subcategory of $[J,C]/D$ whose objects are those morphisms $D' \to D$ of $[J,C]$ where $D' = \Delta_{J,c}$ for some $c$.
If $\mathrm{Cone}(D)$ has a final object, it (the final object) is said to be the universal cone over $D$ or the limit of $D$.

A cartesian product in a category $C$ is the limit of any diagram in $C$ indexed by the discrete category with two objects and no non-identity arrow.