> Not only is this subcategory a preorder, it's also a partial order. And a total order.

I don't think it's a partial order.

Some of the objects in this category are \$$\varnothing, \\{\varnothing\\},\$$ and \$$\\{\\{\varnothing\\}\\}\$$.

There is a unique morphism \$$f : \\{\varnothing\\} \to \\{\\{\varnothing\\}\\}\$$, and another unique morphism \$$g: \\{\\{\varnothing\\}\\} \to \\{\varnothing\\} \$$. So \$$\\{\varnothing\\} \leq \\{\\{\varnothing\\}\\}\$$ and \$$\\{\\{\varnothing\\}\\} \leq \\{\varnothing\\} \$$. But \$$\\{\varnothing\\} \neq \\{\\{\varnothing\\}\\}\$$, so anti-symmetry is violated.

The quotient algebra for this partial order is **Bool**. However, while there is only one [initial object](https://en.wikipedia.org/wiki/Initial_and_terminal_objects) \$$\varnothing\$$ (same as \$$\mathtt{false}\$$ in **Bool**), there is a proper class of singleton sets as final objects rather than just \$$\mathtt{true}\$$.