Morphisms of $\textbf{Set}$ are functions not general relations?

Question

In the category $\textbf{Set}$, where objects are sets and morphisms are functions between sets, why can the morphisms not be any arbitrary relation? The morphisms have to preserve the structure of the objects, but what structure do sets have that functions can preserve but other relations can't? Is it anything to do with there being an infinite number of relations between any two sets while between finite sets there are only a finite number of distinct functions?

If it makes it simpler to not have to consider infinite sets, the same questions still hold for the category of finite sets $\textbf{Set}_{fin}$, where the morphisms are still specificially those relations which are also functions.

Answer 1

You may as well ask, "why is $\mathbb{Q}$ the set of rational numbers rather than the set of real numbers?" The answer is that we just define the symbol $\mathbb{Q}$ to refer to the rational numbers, and instead use the symbol $\mathbb{R}$ to refer to the real numbers.

Similarly, the symbol $\mathbf{Set}$ is simply defined to refer to the category whose objects are sets and morphisms are functions. There's nothing wrong with defining a category whose objects are sets and whose morphisms are relations, but you should give that category a different name since everyone else agrees by convention that morphisms in $\mathbf{Set}$ are functions, and so they will be confused if you use it with a different meaning. The category whose objects are sets and morphisms are relations is traditionally called $\mathbf{Rel}$ instead.

As for why people talk about the category $\mathbf{Set}$ a lot more than they talk about $\mathbf{Rel}$, that is a deeper question. See this MO question for some discussion.

Answer 2

There's nothing in the axioms of what a category is that forces you to use the functions rather than the relations. It's just a matter of definition that we call the category of sets and functions $\mathbf{Set}$. There is also a category of sets and relations, which people call $\mathbf{Rel}$.

Of course there's the question of which of these two categories "deserves" to be considered more fundamental. There's a good argument that $\mathbf{Rel}$ should come first in some sense, since all functions are relations, but not vice-versa. But in everyday mathematical practice functions seem to come up a lot more often than other kinds of relations, so correspondingly $\mathbf{Set}$ gets more attention than $\mathbf{Rel}$.