$\beta \mathbb{N}$ as the free ultrafilter algebra

Applying [L1] and [L3] at $A=\mathbb{N}$ gives the free algebra $(\beta \mathbb{N},{\mu }_{\mathbb{N}})$. Its unit includes every natural, including $0$ and $1$, as the corresponding principal ultrafilter.

Specializing the multiplication formula [L2] to $X=\mathbb{N}$ gives the displayed membership equivalence.

The algebra unit equation ${\mu }_{\mathbb{N}}{\eta }_{\beta \mathbb{N}}=1$ and associativity equation ${\mu }_{\mathbb{N}}\beta ({\mu }_{\mathbb{N}})={\mu }_{\mathbb{N}}{\mu }_{\beta \mathbb{N}}$ are exactly the monad unit and associativity laws in [L3].

Assuming UL/BPI, apply [L4] to extend the cofinite filter on $\mathbb{N}$ to an ultrafilter. It is free: if it were principal at $n$, it would contain both $\{n\}$ and the cofinite set $\mathbb{N}\setminus \{n\}$. Thus $\beta \mathbb{N}$ contains both the principal ultrafilters from step 1.1 and free ultrafilters, but no free ultrafilter is claimed without UL/BPI.