> It is mind blowing for me for some reason like finding out Vader is Luke's father.

Yeah! Even more so: category theory revealed that our _basic concepts in arithmetic and logic_ are related in ways that nobody had suspected before! It's astounding!

> The way they explain it sounds like they are saying that left adjoints are "sum-like" somehow acting like colimits in that they pick out interconnected parts and right adjoints are "product-like" somehow acting like limits in that they find tuples with common traits. Is this a safe intuition to have for adjunctions in general or only true for the examples they use in the book?

It's pretty safe for adjunctions between categories that are fairly similar to \\(\mathbf{Set}\\). You'll noticed that in our study of databases we're focusing on categories of the form \\(\mathbf{Set}^{\mathcal{C}}\\). These are fairly similar to \\(\mathbf{Set}\\) in many ways. (Technically we say they are "toposes".)

If we were working with categories like \\(\mathbf{Set}^{\text{op}}\\), everything would be turned around and your intuitions would be destroyed. A left adjoint functor from \\(\mathcal{C}\\) to \\(\mathcal{D}\\) gives a right adjoint functor from \\(\mathcal{C}^{\text{op}} \\) to \\(\mathcal{D}^\text{op}\\), and vice versa!

Then there are categories like \\(\mathbf{FinVect}\\), the category of finite-dimensional vector spaces and linear maps, that are equivalent to their own opposite. These are neither like \\(\mathbf{Set}\\) nor like \\(\mathbf{Set}^{\text{op}}\\), but somehow poised right in between.

It takes some time to develop intuitions refined enough to handle all these different flavors of category. However, you are on the right track... thinking about important stuff!

Yeah! Even more so: category theory revealed that our _basic concepts in arithmetic and logic_ are related in ways that nobody had suspected before! It's astounding!

> The way they explain it sounds like they are saying that left adjoints are "sum-like" somehow acting like colimits in that they pick out interconnected parts and right adjoints are "product-like" somehow acting like limits in that they find tuples with common traits. Is this a safe intuition to have for adjunctions in general or only true for the examples they use in the book?

It's pretty safe for adjunctions between categories that are fairly similar to \\(\mathbf{Set}\\). You'll noticed that in our study of databases we're focusing on categories of the form \\(\mathbf{Set}^{\mathcal{C}}\\). These are fairly similar to \\(\mathbf{Set}\\) in many ways. (Technically we say they are "toposes".)

If we were working with categories like \\(\mathbf{Set}^{\text{op}}\\), everything would be turned around and your intuitions would be destroyed. A left adjoint functor from \\(\mathcal{C}\\) to \\(\mathcal{D}\\) gives a right adjoint functor from \\(\mathcal{C}^{\text{op}} \\) to \\(\mathcal{D}^\text{op}\\), and vice versa!

Then there are categories like \\(\mathbf{FinVect}\\), the category of finite-dimensional vector spaces and linear maps, that are equivalent to their own opposite. These are neither like \\(\mathbf{Set}\\) nor like \\(\mathbf{Set}^{\text{op}}\\), but somehow poised right in between.

It takes some time to develop intuitions refined enough to handle all these different flavors of category. However, you are on the right track... thinking about important stuff!