I am working on some exercises in Categories for the Working Mathematician and am confused by one of them, which just involves basic examples of functors and natural transformations. It is:
Let $S$ be a fixed set, and $X^s$ the set of all functions $h:S \rightarrow X$. Show that $X \rightarrow X^s$ is the object function of a functor $\bf{Set} \rightarrow \bf{Set}$, and that evaluation $e_x : X^s \times X \rightarrow X$, defined by $e(h,s)=h(s)$, the value of the function $h$ at $s \in S$, is a natural transformation.
What I am having difficulty with is that he only defined the functor on objects, without saying anything about the morphisms. The two conditions for functoriality are based on what the functor does to morphisms. Thus, I do not see how the functor is even defined enough to check either that it is a functor or that $e_x$ is a natural transformation.
Part of the exercise is figuring out what you should define this functor to be on morphisms. The exercise asserts that $X\to X^s$ is "the object function of a functor", which just means there exists some functor with the described properties which sends $X$ to $X^s$ on objects. So you are right that you need to know what it does on morphisms before you can check the rest of the exercise, but part of the exercise is finding a definition on morphisms that will make it work.