Understanding Marsden’s string diagrams for naturality

Question

I'm trying to learn about string diagrams from Dan Marsden's tutorial, and I first get confused on page 9, when he introduces the following two diagrams and claims that they express the naturality condition of a natural transformation $\{\alpha_X:F(X)\to G(X)\;|\;X\in\operatorname{ob}(\mathcal C)\}$.

I understand as follows. The yellow region (left) represents the one-object category $\bf1$, so $X$ and $Y$, which are objects in $\mathcal C$ can be represented as functors from $\bf 1$ to $\mathcal C$, and likewise $f:X\to Y$ as a natural transformation from $X$ to $Y$ (as functors).

What I don't understand is, Why do $\alpha_X$ and $\alpha_Y$ appear on the wire joining $X$ and $Y$? Is this a new notation being introduced (but not explained), or does it follow from the previous discussion in the paper? Also, what "thing" is represented by the wire joining $\alpha_•$ and $f$ in each diagram? My guess is the answers are $X$ and $Y$ respectively, but then again the presence of the $\alpha$'s doesn't seem to make sense.

At this point in the paper, there does not appear to be an explanation of what it means to have four wires connected to a vertex, so maybe that's what I'm missing. But it seems to me that the diagrams would be more correct if we straightened out the wires from $F$ to $G$, shifting the $\alpha$'s to the right. Essentially, I'm looking for a precise explanation of the anatomy of these diagrams.

Answer 1

I’m also not an expert, so take this with a grain of salt.

Let’s first look at something simpler, namely the following picture (from page 13):

This just represents the function $f : X \to G(Y)$. One way of thinking about this is to split the image horizontally into the part above $f$, the part below $f$, and $f$ itself.

• The part above the $f$ (i.e. the yellow region, the green region and the blue region separated by the vertical wires $Y$ and $G$) is just composition of the object $Y$ of $\mathcal D$, thought of as a functor $Y : \mathbf 1 \to \mathcal D$, and $G : \mathcal D \to \mathcal C$. This is a functor $\mathbf 1 \to \mathcal C$, namely the object $G(Y)$ of $\mathcal C$.
• The part below $f$ (i.e. the yellow region and the blue region separated by the vertical wire $X$) is even simpler, it is the object $X$ (again, thought of as a functor $\mathbf 1 \to \mathcal C$.
• Now $f$ (represented by the dot) is a natural transformation from the lower functor to the upper functor. As both of them are simply objects, this is the same thing as a function.

In general, when several wires meet in a dot, this means that the natural transformation represented by the dot is from the composition of the functors represented by the wires entering from below to the composition of the functors represented by the wires entering from above.

In your image, in bottom half of the left diagram we have the function $\alpha_X : F(X) \to G(X)$. In the upper half, we have $f$ but there is $G$ to the right of it, so we have to form the horizontal composition. This is simply $G(f) : G(X) \to G(Y)$. We then form the composition of these two, i.e. $G(f) \circ \alpha_X$ (which is indeed one side of the “usual” equation for natural transformations). Similarly, the other side is $\alpha_Y \circ F(f)$.

As for your question about the wire between $\alpha_X$ and $f$: It is indeed simply $X$ but remember that this wire is not the codomain of $\alpha_X$. The codomain of $\alpha_X$ is the (horizontal) composition of all the things entering from above which are $X$ and $G$; so the codomain is in fact $G(X)$, as expected.