To what extent do functors map "elements" of an object to "elements" of another object? They are usually described as just mapping one object to another.
Thanks
To what extent do functors map "elements" of an object to "elements" of another object? They are usually described as just mapping one object to another.
Thanks
In a concrete category, the definition of concrete category tells you the sense. Even though a faithful functor need not be injective on objects, you can view any morphism as a function between sets by passing to the image of the faithful functor to $\mathrm{Set}$.
Even though your comment mentioned concrete categories, it's important to note that if your category is not concrete, then even when you have a decent way of viewing objects as sets, the morphisms may not be describable in terms of mapping elements. The homotopy category of topological spaces has topological spaces as objects (which have underlying sets), but the morphisms are merely homotopy equivalence classes of continuous functions between underlying sets. These morphisms can't be viewed as mapping particular elements of the underlying sets because different members of the equivalence class would do different things to elements.