I will try to answer the second question.

>Let ${\bf C} \xleftarrow{i} {\cal A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise. Then, the following are equivalent.

>1. $\text{lan}_if \dashv \text{lan}_fi$.
2. $f$ is the $i$-relative left adjoint of $\text{lan}_fi$, i.e. ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, \_ ).$
3. $f = \text{lift}_{\text{lan}_fi}i$ and the lift is absolute.

> If $i$ is only fully faithful $1 \Rightarrow 2$, if $i$ is only dense $2 \Rightarrow 1$.

Proof.

$1 \Rightarrow 2$)  $${\bf B}(f, \_) \stackrel{i \text{ is ff.}}{\cong} {\bf B}((\text{lan}_if) i, \_) \stackrel{1}{\cong} {\bf C}(i, \text{lan}_fi).$$

$2 \Rightarrow 1$)
$${\bf B}(\text{lan}_if, \_) \stackrel{\text{point.}}{\cong} \text{ran}_i{\bf B}(f, \_) \stackrel{2}{\cong} \text{ran}_i{\bf C}(i, \text{lan}_fi) \stackrel{\text{point.}}{\cong} {\bf C}(\text{lan}_ii, \text{lan}_fi)  \stackrel{i \text{ is dense}}{\cong} {\bf C}(\_, \text{lan}_fi).$$

$3$ is just a rewriting of $2$.


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I would like to re-elaborate the $3^{\text{rd}}$ condition. Observe that given a span  ${\bf C} \xleftarrow{i} {\cal A} \xrightarrow{f} {\bf B}$, whe have a couple of maps, $$\text{lan}_{\_}i:[{\bf A},{\bf B}] \leftrightarrows [{\bf B},{\bf C}] :\text{lift}_{\_}i.$$

Are these adjoints?! If so the condition 3. is asking the (co?)unit to be an iso.