I want to make sure I get this...

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-
\\]

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 pullbacks left-adjoint (wasn't that an existential quantifier)?

I'm am getting this right, or am I torturing analogies here?