TITLE: In category theory, why do we define the product by its universal property the way we do?
QUESTION [7 upvotes]: Let $\mathcal{C}$ be a category and let $(X_i)_{i\in I}$ be a family of objects in $\mathcal{C}$. We say that an object $X$ in $\mathcal{C}$ is the product of $(X_i)_{i\in I}$ if we have morphisms $\pi_i:X\longrightarrow X_i$ such that for every object $Y$ in $\mathcal{C}$ and a family of morphisms $f_i:Y\longrightarrow X_i$, there exists a unique $f:Y \longrightarrow X$ such that the following diagram commutes
$$\require{AMScd}
\def\diaguparrow#1{\smash{\raise.6em\rlap{\scriptstyle #1}
   \lower.3em{\mathord{\diagup}}\raise.52em{\!\mathord{\nearrow}}}}
\begin{CD}
&& X\\
& \diaguparrow{f} @VV\pi_iV \\
Y @>>f_i> X_i
\end{CD}$$
This is how I've read we define a product in a given category $\mathcal{C}$. My question is, why do we define this universal property this way and not in any other way? Why is it interesting having this commutative diagram and not any other?
In the category of Sets, I can see how this corresponds to the cartesian product. Indeed, if we take $X=\prod X_i$ and we define $f(y)=(f_i(y))_{i\in I}$ and we take $\pi_i$ as the $i$-th projections, then we have a commutative diagram and the cartesian product becomes the product in Sets. 
However, I still don't see why do we define the product in that way, it looks a bit unnatural to me. Was the definition motivated as a generalization of the cartesian product of sets so we could have a notion of it in other categories? If so, was this the only way to construct such generalization? Why is it important to have the product defined it this way?
It seems to me that by defining this object called product in any category we obtain an object with richer meaning. I wonder if there is any deep intuition about what a product is, or if it is "merely" an object that satisfies such commutative diagram and nothing else.

REPLY [2 votes]: Thoughts are free, and definitions are free. Your question is what is the purpose to define the product by its universal property the way we do.
Let me first correct your definition. The product of $(X_i)_{i\in I}$ of a family of objects of a category $\mathcal{C}$ is not an object having some property. In fact it is an entity consisting of an object $X$ of $\mathcal{C}$ and a family of morphisms $\pi_i: X\longrightarrow X_i$ such that for each object $Y$ and each family of morphisms $f_i:Y\longrightarrow X_i$ there exists a unique $f:Y \longrightarrow X$ such that the following diagram commutes
$$\require{AMScd}
\def\diaguparrow#1{\smash{\raise.6em\rlap{\scriptstyle #1}
   \lower.3em{\mathord{\diagup}}\raise.52em{\!\mathord{\nearrow}}}}
\begin{CD}
&& X\\
& \diaguparrow{f} @VV\pi_iV \\
Y @>>f_i> X_i
\end{CD}$$
The morphisms $\pi_i$ are an integral part of the product, it 
is not enough to require the mere existence of such morphisms having the universal property. You could of course define a product object of the family $(X_i)_{i\in I}$ as an object $X$ for which there exists a family of morphisms $(\pi_i)_{i\in I}$ having the universal property. But then the morphism $f : Y \to X$ would depend on the particular choice of $(\pi_i)_{i\in I}$ and you end up with the above product concept: To find $f$, you need an object plus a family of morphisms.
You see that the purpose is simply to replace a family of morphisms $f_i: Y\longrightarrow X_i$ by a single morphism $f : Y \to X$. Is this reasonable? This is philosophical question. In fact, most definitions in category theory generalize standard concepts in the category of sets. If you think that the cartesian product of sets is a reasonable concept and agree that it is characterized by the above  universal property, then you should also accept that the general definition is not "unnatural". In fact, products in the categorical sense exist in many categories (e.g. in the categories of groups, abelian groups, rings, topological spaces, ...).