Connected component equivalence relation on a Category Hello I have the next question about connected component of one category $C$, they define the next relation for two objects in the category $C$ as follow:
$xRy$ if and only if $Hom_{C}(x,y)\not=\emptyset$
This relation is an equivalence relation. The only problem I have is the symmetric part which I don't see how to prove it, because if I have $f\in Hom_{C}(x,y)$, How do I guarantee that $Hom_{C}(y,x)\not=\emptyset$?
Thank you for your time!!
 A: I don't know who "they" are, but the relation you've given is not an equivalence relation, as you can see by looking at a two-element category with exactly one arrow between the two objects.
It does, however, generate an equivalence relation.  For defining connected components of a category, we should take the smallest equivalence relation containing the one you mentioned.
A: It's not an equivalence relation, since as you observe, there's no reason for it to be symmetric.  For instance, in the category with two objects $x$ and $y$ and only one non-identity morphism $x\to y$, then $xRy$ is true but $yRx$ is not.  The usual definition of connected components of a category is using the equivalence relation generated by your relation $R$ (i.e., the smallest equivalence containing it).  Explicitly, you can show that this is the following equivalence relation: $x$ is equivalent to $y$ iff for some natural number $n$, there exist objects $x_0=x, x_1, \dots, x_n$ and $y_0,y_1,\dots, y_n=y$ and maps $x_0\to y_0\leftarrow x_1 \to y_1\leftarrow x_1\to\dots \to y_{n-1}\leftarrow x_n\to y_n$.
