Category Theory Functors – Natural Transformations Induced by Finite Products

Question

Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane.

Proposition: If a category $C$ has a terminal object $t$ and a product diagram $a\leftarrow a\times b\rightarrow b$ for any two of its objects, then $C$ has all finite products. The product objects provide, by $\langle a, b\rangle\mapsto a\times b$, a bifunctor $C\times C\to C$. For any three objects $\mathbf{a, b}$ and $\mathbf{c}$ there is an isomorphism $\mathbf{\alpha=\alpha_{a, b, c}: a\times(b\times c)\cong(a\times b)\times c}$ natural in $\mathbf{a, b}$ and $\mathbf{c}$.

My Question: I struggle with the boldface part. Take $c$, for example. I suppose for each fixed pair $\langle a, b\rangle$ the natural transformation, called it $\tau$, is between the two functors $S=a\times(b\times-)$ and $T=(a\times b)\times-: C\to C$. The object functions for $S$ and $T$ are obvious: $c\mapsto a\times(b\times c)$ and $c\mapsto(a\times b)\times c$. But what about the arrow functions? And when it comes to $\tau$, each $c$ of $C$ is mapped to which arrow of $C$?

The book can be a bit tough to follow sometimes; but I guess that's part of the charm. Any help would be greatly appreciated.

Answer 1

Some hints.

Let $T:C\times C\to C$ denote product. You then define $a\times(b\times c)$ on arrows and objects very easily as the composite functor $T\circ(1\times T):C\times C\times C\to C\times C\to C$. Similarly, $(a\times b)\times c$ stands for $T\circ(T\times 1)$. Well, that assumes you understand how $T$ is a functor. If the reader is not familiar with this, here's how it works:

For an arrow $(f,g):(x,y)\to(x',y')$ in $C\times C$, $T(f,g)$ - usually written $f\times g$ - is some arrow $x\times y\to x'\times y'$; it is the unique arrow such that $\pi_1 T(f,g)=f\pi_1:x\times y\to x\to x'$ and $\pi_2T(f,g)=g\pi_2:x\times y\to y\to y'$. Such an arrow exists and is unambiguously defined precisely by the universal property of $x'\times y'$.

This definition is an instance of the more general property that $\varprojlim$ is a functor, and $(f,g)$ defines a natural transformation between the diagrams whose limits are $x\times y$ and $x'\times y'$.

To get a map $(a\times b)\times c\to a\times(b\times c)$, use the product universal property in a similar fashion. You just need to find arrows $(a\times b)\times c\to a$ and $(a\times b)\times c\to b\times c$. The only sensible ones you can think of (if you're stuck, what are they for $C=\mathsf{Set}$?) are the correct ones.