> 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 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 left Kan extension. The right adjoint is the coproduct \\( + \mathbf{Set}^2 \to \mathbf{Set}\\), and we saw that this is a right Kan extension.

I'm confused. I thought \\(\Delta \dashv \times\\) ... is that correct? If so I think you mean "the *right* adjoint to \\( \Delta : \mathbf{Set} \to \mathbf{Set}^2 \\) is the product \\( \times : \mathbf{Set}^2 \to \mathbf{Set} \\)".

Moreover, I understand from reading [Bartosz Milewski's blog 2017](https://bartoszmilewski.com/2017/04/17/kan-extensions/) for every functor \\(K : \mathcal{C} \to \mathcal{D}\\) with left and right adjoints:

\[ \mathrm{Ran}\_K\ Id\_{\mathcal{C}} \dashv K \dashv \mathrm{Lan}\_K\ Id\_{\mathcal{C}} \]

But, in the category of functors, this is *flipped*:

\[ \mathrm{Lan}\_K\ (-) \dashv (-) \circ K \dashv \mathrm{Ran}\_K\ (-) \]

This is rather confusing! The proofs of these things have been a bit challenging for me too...

I'm confused. I thought \\(\Delta \dashv \times\\) ... is that correct? If so I think you mean "the *right* adjoint to \\( \Delta : \mathbf{Set} \to \mathbf{Set}^2 \\) is the product \\( \times : \mathbf{Set}^2 \to \mathbf{Set} \\)".

Moreover, I understand from reading [Bartosz Milewski's blog 2017](https://bartoszmilewski.com/2017/04/17/kan-extensions/) for every functor \\(K : \mathcal{C} \to \mathcal{D}\\) with left and right adjoints:

\[ \mathrm{Ran}\_K\ Id\_{\mathcal{C}} \dashv K \dashv \mathrm{Lan}\_K\ Id\_{\mathcal{C}} \]

But, in the category of functors, this is *flipped*:

\[ \mathrm{Lan}\_K\ (-) \dashv (-) \circ K \dashv \mathrm{Ran}\_K\ (-) \]

This is rather confusing! The proofs of these things have been a bit challenging for me too...