> 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}\\).

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}\\).