When we have a category $\mathcal C$, it is usual to define the category $\textrm{Ar}(\mathcal C)$ of morphisms of $\mathcal C$ as the one whose objects are the morphisms of $\mathcal C$ and whose morphisms are commutative squares: If $f:A\to B$ and $g:C\to D$ are elements of $\textrm{Ar}(\mathcal C)$, then a morphism from $f$ to $g$ is a pair of morphisms $(h:A\to C, h':B\to D)$ such that $h'f = gh$.
Therefore, since small categories and functors form a category, it would be expected that the morphisms between functors would be defined this way. But this is not the case, and instead what are used as the morphisms of functors are natural transformations. This is very strange and inelegant: why are morphisms between functors in usual categories defined one way but the ones in the category of categories defined differently?
Note that i'm not asking whether we can define morphisms the way i said, i know we can. I'm also aware of similar questions on this site but none answers where the different treatment of different categories comes from. (And to be honest i don't find any of the anwers in the other related questions satisfying either.)
A morphism between two morphisms only requires two other maps. Morphisms between functors must do that for every morphism in the source category.
If $F:\mathcal C\to\mathcal D$ and $G:\mathcal C\to\mathcal D$ are two functors then we require a commutative diagram for the four objects $F(A), F(B), G(A), G(B)$ for each $A,B\in\mathcal C$ for which a morphism $f:A\to B$ exists.
$\require{AMScd}$ \begin{CD} F(A) @>F(f)>> F(B)\\ @V \eta_A V V @VV \eta_B V\\ G(A) @>>G(f)> G(B) \end{CD}
A commutative diagram as above is required for every morphism in $\mathcal C$. If we imagine a category as a diagram of nodes and arrows, a functor maps the entire diagram to a diagram in another category. The natural transformation ensures that the entire diagrams of $F(\mathcal C)$ and $G(\mathcal C)$ commute.