What defines arrows equality I have decided to learn basics of category theory, but have stumbled upon the very first exercise: given a category C, prove that identity arrow is unique among arrows with domain of X and codomain of X, where X is from the objects of the given category C. But I fail to see or find any definition of arrows equality or inequality. In other words, given 2 arrows: $f:X\to Y $ and $g:X\to Y$, how can I say if they are same or not?
 A: Say you have two identity arrows $\operatorname{Id}_1, \operatorname{Id}_2:X\to X$. By the defining property of identity arrow, we have
$$
\operatorname{Id}_1 = \operatorname{Id}_1\circ \operatorname{Id}_2 = \operatorname{Id}_2
$$
and thus the two are equal.
Basically, the axioms and definitions of your theory will tell you when two things are equal. In this case, an identity arrow $\operatorname{Id}:X\to X$ is defined by the following: for any $f:X\to Y$ and any $g:Z\to X$, we have $f = f\circ{\operatorname{Id}}$ and $g = {\operatorname{Id}}\circ g$. One can deduce general results (usually called theorems) which will assist you in less simple cases so you don't have to appeal directly to the axioms all the time, but in the end, all equalities are proven from whatever equalities your axioms and definitions give you.
Exactly how you should prove that $f$ and $g$ in your question are equal will depend greatly on how they are defined, and what you know about the category in which you are working. Some categories only have one arrow for each (ordered) pair of objects, and in that case, they will automatically be equal. Other categories are more complicated. In most common categories, like the categories of groups (abelian or general), topological spaces, and so on, equality of arrows is not commonly shown on a category theoretical level, although some specific cases can benefit greatly from a category theoretical formulation.
