TITLE: Do opposite categories always exist?
QUESTION [13 upvotes]: The opposite category of C is defined by setting hom(x,y)=hom(y,x) for all objects x,y of C. However in a concrete category, morphisms do not always have inverses (because functions don't always have inverses). So I am wondering how it is possible to always construct a reversed arrow for a given arrow and whether or not it is always possible to construct the opposite category.
Edit: Thank you for the responses and references I think I understand the main part of the question. As I understand it, when you construct the opposite category you forget all the meaning that the objects and morphisms have and reverse the arrows only in terms of classes of objects and sets of morphisms. One question I have is this. When you say a statement about a category you get the dual statement for "free" because the proof also applies in the opposite category. Do you only get things for free when your are talking about types of categories and not specific categories. For example in the category of groups, products always exist but coproduction do not.

REPLY [5 votes]: If $\mathcal{C}$ is a category, then the opposite category $\mathcal{C}^{op}$ is just some other category.  It's not the same category as $\mathcal{C}$.
So if you have any statement $P$ you can prove for all categories, that statement will be valid for $\mathcal{C}^{op}$.  In particular, the statement that $P$ is true for $\mathcal{C}^{op}$ implies the "dual version of $P$" is true for $\mathcal{C}$.  This is how you can dualize theorems "for free" in category theory: if $P$ is true in all categories, then the dual of $P$ is also true in all categories (by applying $P$ to the opposite category).
However, this only works if you know $P$ is true in all categories.  So for instance, in the category $\mathtt{Set}$ it is true that if $X$ is an object and there exists a map from $X$ to an initial object, then $X$ is initial.  (Concretely, if there is a map from $X$ to the empty set, $X$ must be empty.)  The dual statement would be that if $X$ is an object and there exists a map from a terminal object to $X$, then $X$ is terminal.  This statement is not true in $\mathtt{Set}$!  (Indeed, any nonempty set $X$ has a map from a terminal object.)
All that is going on here is that the statement $P$ "if $X$ is an object and there exists a map from $X$ to an initial object, then $X$ is initial" is not true in every category.  So there is no reason to think that if $P$ is true in $\mathcal{C}$, it must also be true in $\mathcal{C}^{op}$.  In particular, while it is true in $\mathtt{Set}$, it is not true in $\mathtt{Set}^{op}$.  As a result, the dual of $P$ is not true in $\mathtt{Set}$.