This is probably wrong or I'm talking funny but I'll try **Puzzle 163** the best I can in hopes of someone giving a solution.

So using John's tip in [comment 27](https://forum.azimuthproject.org/discussion/comment/19814/#Comment_19814) I will define the functors first.

So say we have a category \$$\mathcal{C}, \mathcal{D} \in Ob(\mathbf{Cat})\$$ and a functor \$$F : \mathcal{C} \rightarrow \mathcal{D}\$$. Also \$$x,y \in Ob(\mathcal{C})\$$. We also need \$$S,T \in \mathbf{Set}\$$ and a function \$$f : S \rightarrow T\$$.

**\$$\mathbf{Cat} \stackrel{R_1}{\longrightarrow} \mathbf{Preorder}\$$**

Let \$$[\leq]\$$ be the isomorphism class on the morphisms in \$$homset(x,y)\$$ so that
$$homset(x,y) \iff x \; [\leq] \; y$$

Define \$$R_1(C)\$$ to be the preorder formed by applying **Reflexivity** and **Transivity** using \$$[\leq]\$$.

Define \$$R_1(F)\$$ to be the monotone function such that \$$x \; [\leq] \; y \iff m_1(x) \; [\leq] \; m_1(y)\$$ where \$$m_1\$$ is a monotone function in \$$\mathbf{Preorder}\$$.

**\$$\mathbf{Preorder} \stackrel{R_2}{\longrightarrow} \mathbf{Poset}\$$**

Let \$$[x]\$$ be the isomorphism class on the morphisms in \$$x \in Ob(C)\$$ so that
$$x \; [\leq] \; y \; and \; y \; [\leq] \; x \iff [x] \; = \; [y]$$

Define \$$R_2R_1(C)\$$ to be the preorder formed by applying **Reflexivity** and **Transivity** using \$$[\leq]\$$ and \$$[x]\$$.

Define \$$R_2R_1(F)\$$ to be the monotone function such that \$$[x] \; [\leq] \; [y] \iff m_2([x]) \; [\leq] \; m_2([y])\$$ where \$$m_2\$$ is a monotone function in \$$\mathbf{Poset}\$$.

**\$$\mathbf{Poset} \stackrel{R_3}{\longrightarrow} \mathbf{Set}\$$**

Define \$$R_3R_2R_1(C)\$$ to be the set \$$Ob(\mathcal{C})\$$ such that \$$[x] \; [\leq] \; [y] \iff [x],[y] \in Ob(\mathcal{C})\$$.

Define \$$R_3R_2R_1(F)\$$ to be the function \$$m_3\$$ such that \$$[x] \rightarrow m_2([x])\$$ for all \$$[x] \in Ob(\mathcal{C})\$$

Then we can daisy chain the naturality square Cheuk Man used and proved the naturality for in [comment 39](https://forum.azimuthproject.org/discussion/comment/19830/#Comment_19830) to define the left adjoints.

$\begin{matrix} \mathbf{Cat}(L_1L_2L_3(S), \mathcal{C}) & \overset{\mathbf{Cat}(L_1L_2L_3(\mathbf{f}), \mathbf{F})}\longrightarrow & \mathbf{Cat}(L_1L_2L_3(T), \mathcal{D}) \\\\ \alpha_{L_2L_3(S),\mathcal{C}} \downarrow & & \alpha_{L_2L_3(T),\mathcal{D}} \downarrow \\\\ \mathbf{Pre}(L_2L_3(S), R_1(\mathcal{C})) & \underset{\mathbf{Pre}(L_2L_3(\mathbf{f}), R_1(\mathbf{F}))}\longrightarrow & \mathbf{Pre}(L_2L_3T, R_1(\mathcal{D}))\\\\ \alpha_{L_3(S),R_1(\mathcal{C})} \downarrow & & \alpha_{L_3(T),R_1(\mathcal{D})} \downarrow \\\\ \mathbf{Pos}(L_3(S), R_2R_1(\mathcal{C})) & \underset{\mathbf{Pos}(L_3(\mathbf{f}), R_2R_1(\mathbf{F}))}\longrightarrow & \mathbf{Pos}(L_3(T), R_2R_1(\mathcal{D}))\\\\ \alpha_{S,R_2R_1(\mathcal{C})} \downarrow & & \alpha_{T,R_2R_1(\mathcal{D})} \downarrow \\\\ \mathbf{Set}(S, R_3R_2R_1(\mathcal{C})) & \underset{\mathbf{Set}(\mathbf{f}, R_3R_2R_1(\mathbf{F}))}\longrightarrow & \mathbf{Set}(T, R_3R_2R_1(\mathcal{D}))\\\\ \end{matrix}$