TITLE: What are morphisms of functors
QUESTION [13 upvotes]: I am not been able to understand, what is a morphism between two functors. I have gone through the formal definition involving a commutative diagram. Can someone explain that to me in a bit more details or in pictures?

REPLY [21 votes]: Given categories $\mathcal{C}$ and $\mathcal{D}$, a functor $F:\mathcal{C}\to\mathcal{D}$ consists of two pieces of information:

A rule assigning, to each object $x\in \mathrm{ob}(\mathcal{C})$, a corresponding $Fx\in\mathrm{ob}(\mathcal{D})$.
A rule assigning, to each morphism $f\in\mathrm{Hom}_{\mathcal{C}}(x,y)$, a corresponding $Ff\in\mathrm{Hom}_{\mathcal{D}}(Fx,Fy)$.

"Pictorally":
$${\mathrm{ob}(\mathcal{C})\xrightarrow{F\text{ (on objects)}}\mathrm{ob}(\mathcal{D})\atop x\;\;\;\;\;\;\;\;\to\;\;\;\;\;\;\;\; Fx}\qquad\qquad {\mathrm{Hom}_{\mathcal{C}}(x,y)\xrightarrow{F\text{ (on morphisms)}}\mathrm{Hom}_{\mathcal{D}}(Fx,Fy)\atop \begin{pmatrix} x\\ \downarrow\!\scriptsize f\\ y\end{pmatrix} \;\;\;\;\;\;\;\;\to\;\;\;\;\;\;\;\;\begin{pmatrix} Fx\\ \downarrow\!\scriptsize Ff\\ Fy\end{pmatrix} }$$
Of course, there are also properties that these rules must have, to ensure that we have some consistency: we must have $F(\mathrm{id}_x)=\mathrm{id}_{Fx}$ and $F(g\circ f)=Fg\circ Ff$. We can't just haphazardly move objects and morphisms around without respecting composition.
Now, what would a "morphism of functors" be? We need a sort of "meta-rule"; a rule for turning the rules defining $F$ into the rules defining $G$. 
$$\begin{array}{ccc}
&  & Fx\\
&  \nearrow & \color{red}{\downarrow}\\
x & & \color{red}{\downarrow}\\
& \searrow & \color{red}{\downarrow}\\
& & Gx
\end{array}\qquad\qquad \begin{array}{ccc}
&  & (Fx\xrightarrow{Ff}Fy)\\
&  \nearrow & \color{red}{\downarrow}\\[-0.4in]
(x\xrightarrow{f} y) & & \color{red}{\downarrow}\\
& \searrow & \color{red}{\downarrow}\\
& & (Gx\xrightarrow{Gf}Gy)
\end{array}$$
So, to specify a morphism of functors (let's call it $\alpha$), we can figure that we'll need to do two things:

for each $x\in\mathrm{ob}(\mathcal{C})$, we want to choose a morphism $\alpha_x:Fx\to Gx$
for each $f\in\mathrm{Hom}_\mathcal{C}(x,y)$, we want to choose a "map of arrows" from $Ff$ to $Gf$. What is a map of arrows? Why, it's a choice of arrows on each end that make it a commutative square. And hey, we've already chosen maps $\alpha_x:Fx\to Gx$ and $\alpha_y:Fy\to Gy$, so we actually don't need to be choosing anything more; we just add the requirement that the $\alpha_x$'s we chose make a commutative diagram
$$\begin{array}{ccc}
Fx & \xrightarrow{Ff} & Fy\\
\scriptsize\alpha_x\!\normalsize\downarrow & & \downarrow \!\scriptsize\alpha_y\\
Gx& \xrightarrow[Gf]{} & Gy
\end{array}$$
for any morphism $f$ in the category $\mathcal{C}$.

This choice of arrows $\alpha_x$ for each $x\in\mathrm{ob}(\mathcal{C})$ defines a natural transformation $\alpha$ from $F$ to $G$.

There is actually a very deep connection between the notion of natural transformation, and the notion of homotopy (from the field of topology). In the gif animation below (taken from Wikipedia), what we start with are two different plane curves, $f:X\to \mathbb{R}^2$ and $g:X\to\mathbb{R}^2$, where $X$ stands for some closed interval. A homotopy is a "morphing" of one curve to the other. 
But what does this really mean? For each $s\in X$, we have chosen a path $h_s:I\to \mathbb{R}^2$ that starts at $f(s)$ and ends at $g(s)$ ($I$ stands for the interval $[0,1]$). Using $t$ to denote the variable for these $h_s$'s, you can think of $t$ as time: at time $t=0$, we have $h_s(0)=f(s)$, and at time $t=1$, we have $h_s(1)=g(s)$, and the curve $h_s(t)$ is the path traced out as $f(s)$ moves to $g(s)$. Of course, there must be some consistency; nearby points on one curve can't have too wildly different paths, the maps $h_s$ must "continuously vary" with $s$.

Sounds familiar! In fact, there is a category analogous $I$, let's call it $\mathcal{I}$: it has two objects, named $0$ and $1$, and it has one non-identity morphism $T:0\to 1$. To be poetic, we might call it the "arrow of time". Just as a homotopy from a map $f:X\to Y$ to a map $G:X\to Y$ is a map $h:X\times I\to Y$ such that $h(x,0)$ is $f(x)$, $h(x,1)$ is $g(x)$, and for each $x\in X$, $h(x,t)$ is traces out a  curve from $f(x)$ to $g(x)$, a natural transformation $\alpha$ from a functor $F:\mathcal{C}\to\mathcal{D}$ to a a functor $G:\mathcal{C}\to\mathcal{D}$ can be identified with (or even alternatively defined as) a functor $H:\mathcal{F}\times \mathcal{I}\to\mathcal{G}$, where $H(x,0)=Fx$, and $H(x,1)=Gx$, and $H(x,T)=\alpha_x$.
See this MathOverflow thread Natural transformations as categorical homotopies and this nLab article geometric realization of categories.