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?