I'm reading a few books on category theory, and they talk about a functor from Set to List, with a object s in Set mapped to a list of elements of s. However, there are many lists possible from s, whereas I thought a functor should take one object to one object.


Am I missing something? I'm a programmer, and this functor is important per the books since its the applyall or mapall functor from Set to List, so any help understanding this is very appreciated.



Cannot understand the functor from Set to List
As a functor, applyall should be a functor $L$ from Set to Set (not to some other category List --- I'm not even sure what category is meant by "List"), and it should take any object $s$ of Set (i.e., a set $s$) to the set $L(s)$ of **all** lists of elements of $s$ (not to any particular list). Its action on functions should go like this: If $f:s\to t$ is any morphism in Set (i.e., any function), then $L(f):L(s)\to L(t)$ should be the function which takes any list $(a\_1,a\_2,\dots,a\_k)$ of elements of $s$ to the list $(f(a\_1),f(a\_2),\dots,f(a\_k))$of elements of $t$. That is, $L(f)$ applies $f$ to all the members of a list.


