Keith wrote:

> So, given a functor \\(G: \mathcal{D} \to \mathcal{C}\\), we can create a new functor from that acts by pre-composition, this functor acts backward to the original functor, like pullbacks did, so I'll call it \\(G^*\\),

> \\[
G^* : [\mathcal{C},-] \to [\mathcal{D},-], \\\\
G^* = G\circ-
\\]

You mean

\[ G^* = - \circ G , \]

because as you said, we are _pre_-composing with \\(G\\), and we're using the usual notation where \\((F \circ G)(d) = F(G(d))\\) does \\(G\\) _first_.

> then given another functor \\(F : \mathcal{C} \to \mathbf{Set}\\), we can create a new functor,

> \\[
G^*(F) : [\mathcal{C},\mathbf{Set}] \to [\mathcal{D},\mathbf{Set}],
\\]
so then \\(H :\mathcal{D} \to \mathbf{Set}\\) is sort of like when there was a subset we wanted to look at, and \\(\mathrm{Lan}_G(H)\\) is like the pullback's left-adjoint (wasn't that an existential quantifier)?

> Am I getting this right, or am I torturing analogies here?

No, everything you say here is right!

What I called \\(F \circ G\\) is indeed often called the **pullback of \\(F\\) along \\(G\\)** and written \\(F^\ast(G)\\). The left Kan extension functor \\(\mathrm{Lan}_G\\) is the left adjoint of the pullback functor \\(G^*\\).

And indeed, the left Kan extension is connected to existential quantifiers - though it's more closely connected to sums, which is why Fong and Spivak call it \\(\sum\\). Back when we were studying preorders, also known as \\(\textbf{Bool}\\)-categories, we were often interested in just whether or not something is true. Now we're often interested in the _set of ways_ in which something is true. For example, we don't just care about whether there is a German in our database: we want to know the actual set of Germans!

The only reason I've been writing stuff like

> \[ \textrm{composing with } G : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Set}^{\mathcal{D}} \]

instead of

\[ G^\ast : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Set}^{\mathcal{D}} \]

is that I figure most people are already reeling under the onslaught of new notation and concepts. Congrats on seeing through this!