I came across an exercise in which we want to show an equivalence of categories between a pre-order $P$ and a certain quotient $Q$. It is given that this quotient is induced by the equivalence relation containing the pair $(x,y)$, where $x$ and $y$ are elements of your pre-order, if and only if $x\leq y$ and $y\leq x$.
However, I have some trouble understanding a quotient of a pre-order. The hom-set of two elements in your pre-order consists at most of one arrow. I do not understand how 'dividing out' to some congruence relation changes something to the original structure, since there will be always the same amount of equivalence classes as the original amount of arrows. Most probably, I am wrong here, but I can not yet find the mistake.
Any help or explanation is welcome!
Perhaps a specific example will help. The integers (positive, negative and zero) form a preorder with the relation of divisibility. $m \leq n$ if $m | n$. This is only a preorder and not a partial order since, for example, $1 | -1$ and $-1 | 1$ but $-1 \neq 1$.
The quotient makes these equal to each other. It turns out that $[m] = [n]$ if and only if $|m| = |n|$, so the quotient is equivalent to the natural numbers with ordering given by divisibility.
You might be confused due to the category-theoretic language. If you're considering preorders as categories with category equivalences, then taking this quotient indeed doesn't change anything. The result is equivalent to the original preorder. It is not, however, necessarily isomorphic (in the strict sense), as the above example shows. The actual set of objects may change.