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 of John Bourke, this happens if and only 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.

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 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.