TITLE: Maps in Opposite Categories
QUESTION [27 upvotes]: Given some category ${\mathcal C}$, then the opposite category will consist of the same objects with the morphisms "turned around."  Given $f:A\rightarrow B$, for $A,B$ objects of ${\mathcal C}$, then in general do we have a canonical $f^{op}:B\rightarrow A$ which is induced by $f$?  
Slight Motivation:  This is something that's just been bothering me recently, and I feel like I am thinking about this concept incorrectly.  To make this question slightly less vague, if we had, say, the category of rings and we had the mapping $f:{\mathbb Z}\rightarrow {\mathbb Z}/2{\mathbb Z}$ by $f$ sends even numbers to 0 and odd numbers to 1, then what is the corresponding map in the opposite category, $f^{op}:{\mathbb Z}/2{\mathbb Z}\rightarrow {\mathbb Z}$ induced by $f$?
Also, I always see this $f^{op}$ that I've mentioned just written as $f$; is this shorthand, or is there something deeper there?

REPLY [2 votes]: I thought I'd post this answer in response to the OP's secondary question of when the morphisms of opposite categories are maps of sets. The answer, as it turns out, is that there is a faithful embedding $\mathcal{C}^{\text{op}} \to \textbf{Set}$ if and only if there is a faithful embedding $\mathcal{C} \to \textbf{Set}$, but for stupid reasons.
First, let us consider $\mathbf{Set}^{\text{op}}$. This is a concrete category: consider the contravariant power set functor $\mathcal{P} : \textbf{Set}^{\text{op}} \to \textbf{Set}$. It is representable: $\mathcal{P} \cong \text{Hom}_{\textbf{Set}}(-, 2)$, and is clearly faithful: if $f, g : X \to Y$ are different maps, then for some $x \in X$, $f(x) \ne g(x)$, thus, $x \notin g^{-1}(\{f(x)\})$ but $x \in f^{-1}(\{f(x)\})$. Hence, if there is a faithful embedding $U : \mathcal{C} \to \textbf{Set}$, we can obtain a faithful embedding $$\mathcal{C}^{\text{op}} \xrightarrow{U^{\text{op}}} \textbf{Set}^{\text{op}} \xrightarrow{\mathcal{P}} \textbf{Set}$$
but this embedding is rather ugly and it is sometimes possible to do better, e.g. in the case of integral algebras over an algebraically closed field $k$, the contravariant representable functor $\text{Hom}_{k\textbf{-Alg}}(-, k)$ gives us a faithful embedding of the category of irreducible affine varieties over $k$ into the category of sets.
Moreover, any small category $\mathcal{C}$ is concrete, though again for silly reasons: the Yoneda embedding is a faithful functor $\mathbf{y} : \mathcal{C} \to [\mathcal{C}^{\text{op}}, \textbf{Set}]$, and when $\mathcal{C}$ is small, $[\mathcal{C}^{\text{op}}, \textbf{Set}]$ is concrete. Indeed, if $F : \mathcal{C}^{\text{op}} \to \textbf{Set}$ is a contravariant functor, we just map $F$ to the set
$$\coprod_{c \in \text{Ob}(\mathcal{C})} F(c)$$
and this is obviously faithful, because a natural transformation $F \to G$ is just a collection of maps $F(c) \to G(c)$. Unfortunately, most interesting categories are not small.