Monoids as a category and product structures

Question

I'm currently going through these incredibly awesome lecture notes on category theory and I'm struggling to answer the question posed on page 26 (directly before Lecture 9).

Question: Characterize what it means for a monoid, regarded as a category, to have products. Can a finite monoid have products? I'm using Definition 3 for products, on the bottom of page 22.

Suppose $(M, \times)$ is the monoid that defines the single-object category, I tend to call the object $\cdot$. A product structure will be the $4$-tuple $(\cdot, \pi_1, \pi_2, <-,->)$ where $\pi_1, \pi_2 \in M$ and for every $f_1,f_2 \in M$ I have some element $<f_1,f_2> \in M$ so that $\pi_i \times <f_1,f_2> = f_i$.

I don't see how I could pick such $\pi_i$ unless $M=\{e\}$, in which case $\pi_i=<-,->=e.$ What other $M$ could this work for? Any help would be greatly appreciated.