@John wrote:

> It would be fun to work out necessary and sufficient conditions.

I'm having trouble with this. I've come up with a "hand-waving argument" about sufficient conditions, but can't quite see how to prove things rigorously using just the adjunction definition of the left Kan extension. I thought I'd write out the hand-waving version anyway to check that I'm on the right track more-or-less.

• if \\(G\\) is full then the \\(G\\)-image of \\(\mathcal{D}\\) in \\(\mathcal{C}\\) is "self-contained" in the sense that there are no new arrows going out of the image and coming back in again. That means the "free" construction of \\(\textrm{Lan}_G (H)\\) is well-behaved in the following sense: we are never forced to add new rows to the tables comprising \\(H\\). So \\(\eta_H\\) is surjective.

• if \\(G\\) is faithful then the \\(G\\)-image of \\(\mathcal{D}\\) in \\(\mathcal{C}\\) is "lossless" in the sense that distinct \\(\mathcal{D}\\)-arrows remain distinct in the image. That means the "free" construction of \\(\textrm{Lan}_G (H)\\) is never forces us to identify any rows in the tables comprising \\(H\\). So \\(\eta_H\\) is injective.

Putting these together we get \\(G\\) full and faithful \\(\implies \eta_H\\) bijective \\(\implies \eta\\) iso.