What can be learn about a object $X$ by studying morphisms from $X \to X$ ? I have read at many places that the best way to study about a object $X$ of some category,the "best" way is to study the morphisms from $X \to X$.My question is:

What can be learn about a object $X$  by studying morphisms from $X \to X$ ?

I am mainly interested in the following categories:
$1$ Category of Groups where morphisms are homomorphism 
$2$ Category of Topological Spaces where morphism are Continuous maps
$3$ Category of Vector Spaces where morphisms are Linear maps
$4$ Category of Measurable spaces where morphisms are measurable maps. 
$5$ Category of Lie Algebras where morphisms are Lie maps.
P.S: I don't have any deep idea about category theory,i know the basics of category theory. 
 A: In some sense, this is a very broad question.
Even prior to category theory, some people like to study objects by studying groups acting on those objects. 

For example, the Euclidean group $E(2)$ acts on the Euclidean plane.
It has a normal subgroup $SE(2)$ such that $E(2) / SE(2) \cong \mathbb{Z} / 2 \mathbb{Z}$; this tells us that the Euclidean plane has a notion of orientation. Note that this fact arises simply from the group structure of $E(2)$, without any prior knowledge of the Euclidean plane.
$E(2)$ has a normal subgroup $T(2)$, the "translation group", and $E(2) / T(2) \cong O(2)$, the orthogonal group. This tells us:


*

*The cosets of $O(2)$ are some sort of 'geometric' object that is unchanged by orthogonal transformations

*Between any two such objects, there is a unique element of $T(2)$ that "translates" from one to the other


In other words, the Euclidean plane is made out of points in the familiar way.

Now, I don't think I'd say category theory encourages this at all, aside from providing tools for more doing more sophisticated analyses.
The thing category theory encourages is thinking of the morphisms $Y \to X$  for any $Y$ (or maybe for any $Y$ from a particularly nice subset of objects) as being a good substitute for the notion of "element of $X$" — in fact, it's arguably a better notion of "element" than the ordinary notion of element when one exists.
