I'll kick this off with my solution because this is a fun one to see a bunch of different answers

a) A single object category whose morphisms are the Natural numbers, composition given by \\(x ; y = x * y \\), and the identity morphism on the sole object is \\(0 \\).

This obeys the associative law because \\((x * y) * z = x * (y * z) \\)

This fails to obey the unit law as you can see by the example \\(0 * 1 \neq 1 \\)

b) A single object category whose morphisms are the natural numbers, composition given by \\(x ; y = |x - y| \\), and the identity morphism on the sole object is \\\(0 \\).

This obeys the unit law because \\(|0 - f| = f \\) and \\(|f - 0| = f \\) when \\(f \geq 0 \\), which is true because \\(f \in \mathbb{N} \\)

This fails to obey the associative law as you can see by example \\(||1 - 2| - 3| = 2 \\) and \\(| 1 - | 2 - 3 || = 0 \\)

a) A single object category whose morphisms are the Natural numbers, composition given by \\(x ; y = x * y \\), and the identity morphism on the sole object is \\(0 \\).

This obeys the associative law because \\((x * y) * z = x * (y * z) \\)

This fails to obey the unit law as you can see by the example \\(0 * 1 \neq 1 \\)

b) A single object category whose morphisms are the natural numbers, composition given by \\(x ; y = |x - y| \\), and the identity morphism on the sole object is \\\(0 \\).

This obeys the unit law because \\(|0 - f| = f \\) and \\(|f - 0| = f \\) when \\(f \geq 0 \\), which is true because \\(f \in \mathbb{N} \\)

This fails to obey the associative law as you can see by example \\(||1 - 2| - 3| = 2 \\) and \\(| 1 - | 2 - 3 || = 0 \\)