Opposite category trivial example I have noticed that the basic notion of opposite category puzzles more than one person. I have also read many complete and motivated answers, as well as read  definitions on books by Awodey, MacLane, Walters, etc.
Since I keep on feeling I didn't grasp the very point of it all, I'd like to ask a simple question about the following trivial example.
Let's consider the following sets:
A = {$a_1$, $a_2$}, B = {$b_1$, $b_2$}, and C = {$c_1$, $c_2$}
together with their identity functions and
$f$: A $\to$ B, $g$: A $\to$ C
such that $f$ sends both elements of A into $b_1$ $\in$ B and $g$ sends both elements of A into $c_1$ $\in$ C.
What's the opposite of this category? That is: which elements are contained in A$^{OP}$, B$^{OP}$, and C$^{OP}$, and what do $f^{OP}$ and $g^{OP}$ precisely map?
Thanks in advance.
 A: Your construction involving elements of sets and element-wise definitions of set functions is external to the data defining this as a category. A category consists of only "names" of objects and morphisms (with composition rules), and no other information. In this example the category has objects $\{A, B, C\}$ and morphisms $\{id_A, id_B, id_C, f, g\}$ with the implied compositions. The category doesn't "know" about the elements of the sets, so you wouldn't expect any new category constructed from this one to have a corresponding interpretation for elements. $f^{OP}$, for example, is simply an abstract morphism from $B$ to $A$, and nothing more.
One way in category theory to interpret elements of a set is as morphisms from a singleton set $\{*\}$, so $a_1$ is represented by a morphism from $\{*\}$ to $A$ that sends $*$ to $a_1$. However, passing to the opposite category construction turns these morphisms out of $\{*\}$ into morphisms into $\{*\}$, which no longer have the same interpretation as elements of a set.
A: The opposite category has exactly the same objects and arrows, but the arrows point the other way. So $A^{OP}$ is the same thing as $A$. $f^{OP}$ is a morphism written as $f^{OP}:B^{OP}\rightarrow A^{OP}$. But a morphism in the opposite of this category corresponds to a function in the original category written the opposite way round. So $f^{OP}$ is just the function $f:A\rightarrow B$ written as $f^{OP}:B^{OP}\rightarrow A^{OP}$.
People usually do group theory before category theory so maybe it'll help to have an analogy from group theory. If you have a group $G$ with binary operation $\ast$, then you can define another group $G'$ with binary operator $\otimes$ such that $a\otimes b=b\ast a$. There's nothing new. It's the same group but with multiplications written the other way round. The opposite category is similar: it's the same category with the morphisms written the other way round.
(Actually, the group example is a special case of the opposite of a category. I'll leave that as an exercise.)
