TITLE: What does "Arrows are more important than objects" really mean?
QUESTION [5 upvotes]: I am a final year undergraduate student and I am trying to learn category theory. I am familiar with the basic notions. I am reading Pareigis's notes, http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/04SS/Cats1.pdf. 
In his introduction, he states " Category theory proves that all information about a mathematical object can also be drawn from the knowledge of all structure preserving maps into this object. The knowledge of the maps is equivalent to the knowledge of the interior structure of an object. “Functions are everywhere!” "
I am not sure what he means exactly by this. I have seen before sentences like "It is the arrows that matter most". But I don't see the big picture I guess. Can someone give me some example of how the arrows, in some circumstance, give all the information about the object in a category? (It might be easier for me to understand examples coming from group theory or topology.)

REPLY [2 votes]: There are many different answer to this question I'm gonna write some of the them.
For start we can observe that when dealing with structured set (for instance groups or topological spaces) lot's of the set theoretical structure can be recovered respectively via group homomorphisms and contiuous functions.
For instance there's a one-on-one correspondence between the elements of  (the underlying set of) a group $G$ and the homomorphisms of the type $\mathbb Z \to G$.
Similarly there's a one-on-one correspondence between the points of a space $X$ (i.e. the elements of the underlying set) and continuous function of the form $\{*\} \to X$ (where $\{*\}$ is the pointed space with the obvious topology).
So we can indentify elements of the underlying sets of these structure with the some arrows in their respective categories, meaning that we can recover the element-structure from arrows.
Another example (which is still a topological one is the following): consider the topological space $\{0,1\}$ with the topology $\{\emptyset,\{0,1\},\{1\}\}$ and let's call this space $\Omega$. There's a one-on-one correspondence between continuous functions of type $X \to \Omega$ and open sets of $X$ (for every $f \colon X \to \Omega$ we have the open set $f^{-1}(\{1\})$ and for every $A \subseteq X$ open we have the continuous function $f_A \colon X \to \Omega$ sending every $x \in A$ in $1$ and the other points in $0$).
So again we could recover some internal structure of the topological space (the open sets) in an arrow theoretic fashion.
Finally there's the above mentioned yoneda lemma which proves that morphisms/arrows are what really matters. 
In every category $\mathbf C$ for every $c \in \mathbf C$ there is a canoical functor $\mathbf C(-,c) \colon \mathbf C^\text{op} \to \mathbf {Set}$.
This functor associate to every $x \in \mathbf C$ the set of morphism $\mathbf C(x,c)$. Yoneda lemma states that for every other $c' \in \mathbf C$  the two functor $\mathbf C(-,c)$ and $\mathbf C(-,c')$ are naturally isomorphic (i.e. there's a family of natural bijections $\langle \mathbf C(x,c) \to \mathbf C(x,c')\rangle_{x \in \mathbf C}$)
if and only if $c$ and $c'$ are isomorphic as objects of $\mathbf C$.
This says that the structure (up to isomorphism) of the object of a category can be recovered by the families of sets $\langle \mathbf C(x,c)\rangle_{x \in \mathbf C}$, so by the arrows on that objects.
Of course there are other good example of how to recover information of an object from the arrows but I hope this ones could be sufficient to motivate the motto morphisms are what matters the most.