> **Puzzle 163.** Show there are functors

> \[ \mathbf{Cat} \stackrel{R_1}{\longrightarrow} \mathbf{Preord} \stackrel{R_2}{\longrightarrow} \mathbf{Poset} \stackrel{R_3}{\longrightarrow} \mathbf{Set} \]

> with left adjoints going back:

> \[ \mathbf{Set} \stackrel{L_3}{\longrightarrow} \mathbf{Poset} \stackrel{L_2}{\longrightarrow} \mathbf{Preord} \stackrel{L_1}{\longrightarrow} \mathbf{Cat} . \]

> Hint: the composite \\( R_3 R_2 R_1 : \mathbf{Cat} \to \mathbf{Set} \\) is our friend

> \[ \text{Ob}: \mathbf{Cat} \to \mathbf{Set} ,\]

> and the composite \\(L_1 L_2 L_3 : \mathbf{Set} \to \mathbf{Cat} \\) is our friend

> \[ \mathrm{Disc} : \mathbf{Set} \to \mathbf{Cat} .\]


Wait, doesn't going to Poset skeletonise things?
I.e. any connected* groupoid is taken to the poset \\(\\{\ast\\}\\)? So that the composition doesn't equal Ob?