@Matthew – cheers for that, I suspected the answer was no.

The adjunction tells us that there's a unit natural transformation \$$\eta_H : H \Rightarrow \textrm{Lan}_G(H) \circ G\$$ but there doesn't seem to be in general any reason to think this is a natural isomorphism, let alone an identity of functors.

@Keith – maybe I'm misunderstanding you but it seems to me that this particular example we do have \$$\text{Lan}\_{G} (H) \circ G = H\$$ "on the nose" – the two functors agree on the single object of \$$\mathcal{D}\$$, do they not?