I wouldn't say that \\(\text{Lan}\_{G} (H) \circ G = H\\) on the nose, but rather, \\(\text{Lan}\_{G} (H) \circ G \sim H\\).

In fact, \\(\text{Lan}\_{G} (H) \\) is a sort of pushforward operation, taking \\(H : \mathcal{D} \to \mathbf{Set}\\) and \\(\beta\_\text{Germans} : H \to F\\), then pushing those forward along \\(G : \mathcal{D} \to {C}\\).

\\(H\\) here is acting like a subset of \\(F\circ G\\) (in fact \\(H\\) is a subfunctor of \\(F \circ G\\)) with \\(\beta_\text{Germans}\\) a sort of inclusion natural transformation, and \\( \textrm{Lan}\_{G} (H)\\) is a pushforward of sorts along \\(G\\).