>In general, a \$$\mathbf{Cost}\$$-enriched profunctor \$$\Phi : C \nrightarrow D\$$ is defined to be a \$$\mathbf{Cost}\$$-enriched functor

>$\Phi : C^{\text{op}} \times D \to \mathbf{Cost}$

I am having a hard time seeing how \$$\Phi\$$ is a \$$\mathbf{Cost}\$$-enriched functor. From my understanding, a \$$\mathbf{Cost}\$$-enriched functor, would be a functor going between two \$$\mathbf{Cost}\$$-enriched categories \$$C\$$ and \$$D\$$. But according to the definition this would require the functor to be a monotone function. But these profunctors don't seem to be monotone nor are they functions (at least the example in the lecture is not)...

So looking at the second definition, \$$\Phi : C^{\text{op}} \times D \to \mathbf{Cost} \$$, we are taking the product of two \$$\mathbf{Cost}\$$-enriched categories and sending them to \$$\mathbf{Cost}\$$, the base of enrichment itself. Can the base of enrichment be an enriched category?