I am looking for either (1) confirmation that the following is true, (2) the mistake making it false pointed out to me:
Let $F:\mathcal{C} \to \mathcal{D}$ be a functor from a small category $\mathcal{C}$ into a small category $\mathcal{D}$.
Then the object part of the functor, $F_{obj}$, associates with each object $c \in \mathcal{Ob(C)}$ an object $Fc \in \mathcal{Ob(D)}$, making it a function $F_{obj}:\mathcal{Ob(C)}\to \mathcal{Ob(D)}$ (with both the domain and codomain being sets since $\mathcal{C}$ and $\mathcal{D}$ are small categories).
Likewise, the morphism part of the functor, $F_{mor}$, associates with each morphism $f \in \operatorname{Hom}_{\mathcal{C}}(c, c')$ a morphism $Ff \in \operatorname{Hom}_{\mathcal{D}}(Fc, Fc')$, making $F_{mor}$ a family of functions $\{F_i: \operatorname{Hom}_{\mathcal{C}}(c,c')\to\operatorname{Hom}_{\mathcal{D}}(Fc,Fc')\ |\ (c,c')=i \in \mathcal{C}\times\mathcal{C} \}$.
Hence, ignoring the foundational problem of defining "functions" between proper classes as opposed to sets (i.e. the case when either $\mathcal{C}$ or $\mathcal{D}$ is not small), functors can be thought of as actual (families of) functions, not just arbitrary morphisms between categories.
morphism between categories - consider a category $\mathscr{C}$, for which the objects $\mathcal{C} \in\mathcal{Ob}(\mathscr{C})$ are themselves categories. Then the morphisms of the category $\mathscr{C}$ are "morphisms between categories".
Question: Given a category $\mathscr{C}$ of the type described above, are its morphisms always functors?
I believe not, since the structure described above for functors seems to be more than is strictly necessary to satisfy the category axioms (associativity, identity, closure under composition).