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.