What follows is kind of preliminary and complementary to Ivan's answer.

Let $A$ be a small category, $\mathcal B$ and $\mathcal C$ be locally presentable categories
and $i\colon A \to \mathcal B$ and $j \colon A \to \mathcal C$ two functors.
The left Kan extension $\operatorname{Lan}_i j \colon \mathcal B \to \mathcal C$ is a left adjoint
functor if and only if it commutes with colimits. By [this answer][1] of John Bourke, this happens
if, for any object $a$ of $A$, the object $i(a)$ of $\mathcal B$ is tiny. Suppose that
this is the case and call $V\colon \mathcal C \to \mathcal B$ the right adjoint to
$\operatorname{Lan}_i j$.

If the functor $i \colon A \to \mathcal B$ is fully faithful, then for every object $a$ of $A$ we have
$\operatorname{Lan}_i j \,(i(a)) \cong j(a)$. Thus for every object $a$ of $A$ and every object
$Y$ of $\mathcal Y$ we get
$$\text{Hom}_{\mathcal C}(\operatorname{Lan}_i j \,(i(a)), Y) = \text{Hom}_{\mathcal C}(j(a), Y)
= \text{Hom}_{\mathcal B}(i(a), V(Y)) $$
naturally in $a \in A$. This is equivalent to say that the two commas $A/Y$ and $A/V(Y)$
(aka $j \downarrow Y$ and $i\downarrow V(Y)$) are isomorphic categories.

In the case where $i$ is fully faithful, the functor $V$ is isomorphic to $\operatorname{Lan}_j i$
if and only if for any object $Y$ of $\mathcal C$ we have that $V(Y)$ is the colimit of the functor
$A/Y \to A \to \mathcal B$. By the previous paragraph, this is isomorphic to the colimit of the
functor $A/V(Y) \to A \to \mathcal B$. Thus $V \cong \operatorname{Lan}_j i$ if and only if every
object in the image of $V$ is canonically a conical colimit with respect to the functor $i$. (The
functor $i$ is dense in the subcategory of $\mathcal B$ spanned by the essential image of $V$.)

We obtain a sufficient condition for $V$ to be isomorphic to $\operatorname{Lan}_j i$ imposing the functor $i$ to be dense (and fully faithful). Indeed, in this case by definition we have that $V(Y)$ is the colimit of the functor $A/V(Y) \to A \to \mathcal B$ for any object $Y$ of $\mathcal C$ and so we conclude using the preceding paragraph. 

  [1]: https://mathoverflow.net/a/112206/42137