> **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

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


Has anyone got anywhere on **Puzzle 163**? I'm thoroughly stuck.

The inclusion functors **Poset** → **Preord** and **Preord** → **Cat** have _left_ adjoints: we can send a preorder to its skeleton, and a category to its "thin-ification", but I don't think they have right adjoints, and I can't think of any other sensible functors between these categories.

The only bit I can get is to work is the discrete functor **Set** → **Poset**, which has a right adjoint: send a poset to its underlying set.

Also the underlying functor **Preord** → **Set** has a left and a right adjoint (the discrete and codiscrete preorders on a set). But this doesn't seem to factor through **Poset** since the codiscrete preorder on a set is typically not a poset.