What's the definition of morphism in a dual category?

Question

$ \newcommand{\cat}{\mathbf} \newcommand{\RR}{\mathrm R} $

If there exist a morphism $$f:\RR \rightarrow [0,+\infty] ,f : x\mapsto x^2+1$$ in a category $\cat{Set}$. In particular, $f$ is not injective nor a surjective.

In the category of $\cat{Set^{op}}$, the $f$ change into $$f^{op}:[0,+\infty]\rightarrow \RR $$

This raises a question : $$f^{op}:x\mapsto ??? $$

The definition must satisfy that: $$ \cat {(C^{op})^{op}} = \cat C $$

Answer 1

The morphism $f^{\mathrm{op}}$ is still the same function as $f$. In a category, the morphisms can be anything; they don't need to be functions. (For example, think about treating a poset like a category: there, the morphisms aren't functions either.) In particular, it is perfectly allowed to make up a category $\mathsf C$, where the objects of $\mathsf C$ are sets, and a morphism $f: X \to Y$ is actually a function $f: Y \to X$. (Can you see what composition has to be?) Then, the category $\mathsf C$ is just $\mathsf{Set}^{\mathrm{op}}$.

Answer 2

In $\mathbf{Set}^\text{op}$, $\hom(X, Y)$ is the set of all functions from $Y$ to $X$.

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That said, there is an equivalent description: that $\hom(X,Y)$ is the set of all functions $\mathcal{P}(X) \to \mathcal{P}(Y)$ that preserve unions and intersections.

More precisely, $\mathbf{Set}^\text{op}$ is equivalent to the category whose objects are complete atomic boolean algebras and whose morphisms are functions that preserve all unions and intersections.

The bijection between the two descriptions associates to a function $f : Y \to X$ the function $f^* : \mathcal{P}(X) \to \mathcal{P}(Y)$ defined by $$f^*(S) = \{ y \in Y \mid f(y) \in S \} $$

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It may be interesting to note that there are naturally occurring categories where morphisms involve functions going the "wrong way". For example, in Top, the category of topological spaces and continuous maps, the homomorphisms $(|X|, \mathcal{S}) \to (|Y|, \mathcal{T})$ involve functions $|X| \to |Y|$ (i.e. a mapping on points) and also functions $\mathcal{T} \to \mathcal{S}$ (the inverse image of an open set is open).

A related notion is that of a locale. It's like a topological space, but without the set of points: a locale is just the frame of open 'sets'. So, in the category Loc of locales, a locale morphism $\mathcal{S} \to \mathcal{T}$ is a (structure preserving) function $\mathcal{T} \to \mathcal{S}$!