Explain Mac Lane, CTFWM, page 109

Question

In Category Theory for the Working Mathematician, second edition, page 109 (near the bottom of the page) is said

but all triangles formed by these projections need not commute ($f_n p_{n+1} \ne p_n$).

Please explain why they need not commute.

Answer 1

Notation: $0=\varnothing$, $1=\{\varnothing\}$, $2=\{\varnothing,\{\varnothing\}\}$.

Let's define the functor $F\colon\omega^{op}\to\mathbf{Set}$ in the following way: $$F_i=\begin{cases} 2,i=0\\ 1,i\ne0 \end{cases};$$ $$ f_i=\begin{cases} \Delta_0,i=0\\ id(1),i\ne0 \end{cases}.$$ Then $(f_0\circ p_1)(1,0,0,\ldots)=0\ne 1=p_0(1,0,0,\ldots)$.

Answer 2

Because it is a product. Product is taken not on the cone $F$ but on the cone corresponding to (discrete) set $\omega$. Thus equalities related to the cone $F$ are not required.