Proposition 1.3.2 in Handbook of Categorical Algebra Volume I states that if $\mathcal{C}$ is a small category and $\mathcal{D}$ is a category, then the functors from $\mathcal{C}$ to $\mathcal{D}$ and their natural transformations form a category.
I am trying to show this. I believe I need to show that there exists a class $A$ such that $F \in A$ iff $F$ is a functor from $\mathcal{C}$ to $\mathcal{D}$. I think this more or less reduces to showing the following statement:
Let $A$, $B$ be classes and $f:A \rightarrow B$ a class function. If $A$ is a set, then $f(A)$ is a set.
How can I show this statement is true?