"Nice proof" that the unit of the left Kan extension of $F$ is an isomorphism, if $F$ is fully faithful

There is a useful trick for dealing with this. The following appears as Lemma 1.3 in [Johnstone and Moerdijk, Local maps of toposes].

Proposition. Given an adjunction $$L \dashv R : \mathcal{D} \to \mathcal{C}$$ if $\mathrm{id}_{\mathcal{C}} \cong R L$ (as functors) then the unit $\eta : \mathrm{id}_{\mathcal{C}} \Rightarrow R L$ is (also) a natural isomorphism.

Proof. Let $\mu = R \epsilon L$, where $\epsilon : L R \Rightarrow \mathrm{id}_{\mathcal{D}}$ is the counit. Then (by the triangle identities), we have a monad. We can transport this structure along any natural isomorphism $\theta : \mathrm{id}_{\mathcal{D}} \Rightarrow R L$, so that e.g. $$\begin{array}{rcl} \mathrm{id}_{\mathcal{C}} & \overset{\theta}{\to} & \mathrm{id}_{\mathcal{C}} \\ {\scriptstyle \eta} \downarrow & & \downarrow {\scriptstyle \eta'} \\ \mathrm{id}_{\mathcal{C}} & \underset{\theta}{\to} & R L \end{array}$$ commutes. But (using naturality) any comonad structure $(\eta', \mu')$ on $\mathrm{id}_{\mathcal{C}}$ must consist of natural isomorphisms, so we deduce that the original $\eta$ and $\mu$ are also natural isomorphisms.　◼