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I understand that an arrow is between two objects, a functor is between two categories. And then a natural transformation is, according to Goldblatt's Topoi, a comparison of two functors.

Here is Goldblatt's definition:

A natural transformation from functor $F: \mathscr C \to \mathscr D$ to functor $G: \mathscr C \to \mathscr D$ is an assignment $\tau$ that provides, for each $\mathscr C$-object $a$, a $\mathscr D$-arrow $\tau_a :F(a) \to G(a)$, such that for any $\mathscr C$-arrow $f:a\to b$, the diagram commutes in $\mathscr D$, ie. $\tau_b \circ F(f)=G(f)\circ \tau_a$. We use the symbolism $\tau: F \xrightarrow{\tau} G$ to denote that $\tau$ is a natural transformation from $F$ to $G$.

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I have absolutely no idea what's going on. I tried to understand the diagram, but I can't even understand where the $F(a)/F(b)$ and $F(f)$ are coming from. (Is this similar to how structures are preserved under homomorphism? Because the way the composition is formulated looks a bit similar)

I think I am lacking an intuitive understanding of what it means to compare two functors.

Could anyone help please?

It's been pointed out that another question has already covered what I am asking (What intuitive notion is formalized by "natural transformation" in category theory?); the question does also ask for an intuitive perspective on natural transformation.

However it's framed around topology and other mathematical structures, but I really would like to avoid those because coming from a non-mathematical background, framing it around those does not really help (I have no idea what those structures are). If possible, I would like to keep any other structure away from the explanation.

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Constantly confused
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    $F$ is one of the two functors - the other is $G$. Remember that functors act on both objects and arrows. "$F(a)$" and "$F(b)$" are the objects in $\mathcal{D}$ which $F$ sends $a$ and $b$ to, respectively, and "$F(f)$" is the arrow in $\mathcal{D}$ that $F$ sends $f$ to. So our initial idea is that, in order to compare $F$ and $G$, we should begin by trying to relate (via the $\tau$s) what they do to a single arrow $f$ (and when we're looking at an arrow $f$ we should also pay attention to its source $a$ and target $b$). – Noah Schweber Oct 15 '19 at 15:06
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    $F: \mathscr C \to \mathscr D$ is a functor from categories, where a category is "made of" objects and arrows. $a$ is a $\mathscr C$-object and $f$ a $\mathscr C$-arrow. Thus, also $b$ is $\mathscr C$-object. – Mauro ALLEGRANZA Oct 15 '19 at 15:07
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    $F(a)$ is the "image" of $a$ with the functor $F$, and thus is an object of $\mathscr D$, because $F$ maps $\mathscr C$ in $\mathscr D$. The same for $F(b)$. – Mauro ALLEGRANZA Oct 15 '19 at 15:09
  • @ArnaudD. The question does also ask for an intuitive perspective on natural transformation. However it's framed around topology and other mathematical structures, but I really would like to avoid those because coming from a non-mathematical background, framing it around those does not really help. If possible, I would like to keep any other structure away from the explanation. – Constantly confused Oct 15 '19 at 15:15
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    But $F$ is defined also on arrows; thus $a,b$ and $f$ are in $\mathscr C$ while $F(a),F(b)$ and $F(f)$ are in $\mathscr D$. The same for $G$. – Mauro ALLEGRANZA Oct 15 '19 at 15:27
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    Now consider $\mathscr D$ : we have objects $F(a)$ and $G(a)$ and thus the "natural transf" $\tau_a$ is an arrow in $\mathscr D$ that maps $F(a)$ to $G(a)$. – Mauro ALLEGRANZA Oct 15 '19 at 15:29
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    @DanielMak "I really would like to avoid those because coming from a non-mathematical background, framing it around those does not really help. If possible, I would like to keep any other structure away from the explanation." I don't think that category theory by and large is accessible without some significant background in other mathematical topics - specifically, at least some algebra and/or topology. So I'm not sure that's a realistic goal. – Noah Schweber Oct 15 '19 at 18:59

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A natural transformation is between two functors, just as a functor is between two categories, i.e. they are simply arrows in a suitable category. The word 'comparison' has no special meaning here.

Note that in Goldblatt's definition, $F$ and $G$ are given functors, both $\mathscr C\to\mathscr D$, so that they already assign the object $F(a)$ [resp. $G(a)$] of $\mathscr D$ to any object $a$ of $\mathscr C$.
Also, they assign arrows $\underset{a\to b}f\ \mapsto \underset{F(a)\to F(b)}{F(f)}$ [resp. $\underset{G(a)\to G(b)}{G(f)}$].

We can also view natural transformations simply as functors $\mathscr C\to(\mathscr D^\to)$ where $\mathscr D^\to$ is the arrow category of $\mathscr D$, whose objects are the arrows of $\mathscr D$, and a morphism $\underset{a\to b}f\ \to \ \underset{a'\to b'}g$ is defined to be a commutative square $$\matrix{a&\overset f\to & b\\ \downarrow & & \downarrow \\ a' & \underset g\to & b'}$$ In that setup, i.e. if a functor $\varphi:\mathscr C\to (\mathscr D^\to)$ is given, we can recover $F$ and $G$ by composing $\varphi$ with the domain [resp. the codomain] functor $(\mathscr D^\to) \to\mathscr D$, which assigns its arrow on the left [resp. right] to a commutative square.

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