I have a puzzle that I'm not sure how to answer.

Back in [Lecture 57](https://forum.azimuthproject.org/discussion/2281/lecture-57-chapter-4-feasibility-relations) we saw how there's connection between monotone functions and feasibility relations:

> Suppose \\(X\\) and \\(Y\\) are preorders.

>

> **Puzzle 173.** Suppose \\(f : X \to Y \\) is a monotone function from \\(X\\) to \\(Y\\). Prove that there is a feasibility relation \\(\Phi : X \nrightarrow Y\\) given by \\(\Phi(x,y) \iff f(x) \le y\\).

>

> **Puzzle 174.** Suppose \\(g: Y \to X \\) is a monotone function from \\(Y\\) to \\(X\\). Prove that there is a feasibility relation \\(\Psi : X \nrightarrow Y\\) given by \\(\Psi(x,y) \iff x \le g(y)\\).

>

> **Puzzle 175.** Suppose \\(f : X \to Y\\) and \\(g : Y \to X\\) are monotone functions, and use them to build feasibility relations \\(\Phi\\) and \\(\Psi\\). When is \\(\Phi = \Psi\\)?

How do we generalise this to get a connection between \\(\mathcal{V}\\)-functors and \\(\mathcal{V}\\)-profunctors?

Back in [Lecture 57](https://forum.azimuthproject.org/discussion/2281/lecture-57-chapter-4-feasibility-relations) we saw how there's connection between monotone functions and feasibility relations:

> Suppose \\(X\\) and \\(Y\\) are preorders.

>

> **Puzzle 173.** Suppose \\(f : X \to Y \\) is a monotone function from \\(X\\) to \\(Y\\). Prove that there is a feasibility relation \\(\Phi : X \nrightarrow Y\\) given by \\(\Phi(x,y) \iff f(x) \le y\\).

>

> **Puzzle 174.** Suppose \\(g: Y \to X \\) is a monotone function from \\(Y\\) to \\(X\\). Prove that there is a feasibility relation \\(\Psi : X \nrightarrow Y\\) given by \\(\Psi(x,y) \iff x \le g(y)\\).

>

> **Puzzle 175.** Suppose \\(f : X \to Y\\) and \\(g : Y \to X\\) are monotone functions, and use them to build feasibility relations \\(\Phi\\) and \\(\Psi\\). When is \\(\Phi = \Psi\\)?

How do we generalise this to get a connection between \\(\mathcal{V}\\)-functors and \\(\mathcal{V}\\)-profunctors?