>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?

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

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?