@ Christopher

You said

> I think you just systematically replace Bool with V (, and replace an equality with an equivalence, I think).
>
> > Suppose \$$X\$$ and \$$Y\$$ are \$$\mathcal{V}\$$-categories.
>>
>> **Puzzle 173V.** Suppose \$$f : X \to Y \$$ is an \$$\mathcal{V}\$$-functor from \$$X\$$ to \$$Y\$$. Prove that there is a \$$\mathcal{V}\$$-profunctor \$$\Phi : X \nrightarrow Y\$$ given by \$$\Phi(x,y) = Hom(f(x),\,y) \$$. (i.e. that \$$\Phi\$$ is a valid \$$\mathcal{V}\$$-profunctor)
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>> **Puzzle 174.** Suppose \$$g: Y \to X \$$ is a \$$\mathcal{V}\$$-functor from \$$Y\$$ to \$$X\$$. Prove that there is a \$$\mathcal{V}\$$-profunctor \$$\Psi : X \nrightarrow Y\$$ given by \$$\Psi(x,y) = Hom(x,\,g(y))\$$.
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>> **Puzzle 175.** Suppose \$$f : X \to Y\$$ and \$$g : Y \to X\$$ are \$$\mathcal{V}\$$-functors, and use them to build \$$\mathcal{V}\$$-profunctor \$$\Phi\$$ and \$$\Psi\$$. When is \$$\Phi \equiv \Psi\$$?

I think you meant to address Anindya, not me.

My answer to this puzzle differs from yours. We aren't in ordinary category theory so we don't have \$$\mathrm{Hom}\$$ at our disposal.

We do have \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$, which are like \$$\mathrm{Hom}\$$-sets, so I used those.

Moreover I assumed \$$\mathcal{V}\$$ was closed and commutative so I could work with \$$\multimap\$$, because otherwise I don't know how to make \$$\Phi\$$ into a \$$\mathcal{V}\$$-profunctor.

But I have been playing around with a puzzle of my own: How do we define \$$\mathcal{V}\$$-profunctor composition? Can we make \$$\multimap\$$ into a \$$\mathcal{V}\$$-profunctor? What happens if we compose with \$$\multimap\$$?