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\$$.