Matthew - I'd guess left Kan extensions.

First, remember that left adjoints and in particular left Kan extensions tend to like adding a bunch of stuff up, while right adjoints and in particular right Kan extensions act like multiplication. For example, the left adjoint is the coproduct \\( + \mathbf{Set}^2 \to \mathbf{Set}\\), and we saw that this is a left Kan extension. The right adjoint to \\( \Delta : \mathbf{Set} \to \mathbf{Set}^2 \\) is the product \\( \times : \mathbf{Set}^2 \to \mathbf{Set} \\), and we saw in [Lecture 54](https://forum.azimuthproject.org/discussion/2277/lecture-54-chapter-3-tying-up-loose-ends/p1) that this is a right Kan extension.

Second, we'll see that composing profunctors is like matrix multiplication. I'll explain this better later on, but:

An enriched profunctor \\(\Phi : \mathcal{X}^{\text{op}} \times \mathcal{Y} \to \mathcal{V}\\) is like a 'matrix' that hands us an element \\(\Phi(x,y) \\) for each \\(x\\) and \\(y\\). To compose, we just generalize matrix multiplication. As you know, matrix multiplication involves multiplying matrix entries and then summing them up. But we use the monoidal structure in \\(\mathcal{V}\\) to multiply the matrix entries and then a colimit to sum them up. The colimit is the part where Kan extensions might enter... and by my earlier remarks, it would be a left Kan extension.

Note however that it'll be an _enriched_ left Kan extension. If we were looking at ordinary profunctors, which are \\(\mathbf{Set}\\)-enriched profunctors, then we'd be using an ordinary left Kan extension.

First, remember that left adjoints and in particular left Kan extensions tend to like adding a bunch of stuff up, while right adjoints and in particular right Kan extensions act like multiplication. For example, the left adjoint is the coproduct \\( + \mathbf{Set}^2 \to \mathbf{Set}\\), and we saw that this is a left Kan extension. The right adjoint to \\( \Delta : \mathbf{Set} \to \mathbf{Set}^2 \\) is the product \\( \times : \mathbf{Set}^2 \to \mathbf{Set} \\), and we saw in [Lecture 54](https://forum.azimuthproject.org/discussion/2277/lecture-54-chapter-3-tying-up-loose-ends/p1) that this is a right Kan extension.

Second, we'll see that composing profunctors is like matrix multiplication. I'll explain this better later on, but:

An enriched profunctor \\(\Phi : \mathcal{X}^{\text{op}} \times \mathcal{Y} \to \mathcal{V}\\) is like a 'matrix' that hands us an element \\(\Phi(x,y) \\) for each \\(x\\) and \\(y\\). To compose, we just generalize matrix multiplication. As you know, matrix multiplication involves multiplying matrix entries and then summing them up. But we use the monoidal structure in \\(\mathcal{V}\\) to multiply the matrix entries and then a colimit to sum them up. The colimit is the part where Kan extensions might enter... and by my earlier remarks, it would be a left Kan extension.

Note however that it'll be an _enriched_ left Kan extension. If we were looking at ordinary profunctors, which are \\(\mathbf{Set}\\)-enriched profunctors, then we'd be using an ordinary left Kan extension.