@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.