## Lurie's approach

First let me explain why this is already in HTT 4.3.3.

Recall that a (pointwise) left Kan extension of $F: A \to \mathcal{C}$ along an inclusion $i: A \to B$ is a functor $F: B \to \mathcal{C}$ such that, for each $b \in B$, $F(b)$ is a colimit of the diagram $A \times_B B_{/b} \to A \to \mathcal{C}$.

In general, if $g: A \to B$ is a functor between $\infty$-categories, and $\mathcal{C}$ another $\infty$-category. Then a left Kan extension of $F: A \to \mathcal{C}$ along $g$ is a functor $\overline{F}: (A\times [1]) \amalg_{A \times \{1\}} B \to \mathcal{C}$ which is a left Kan extension (in the previous sense) of its restriction to $A = A \times \{0\}$. (Notice that the data of $\overline{F}$ is precisely the data of a functor out of $B$ and a natural transformation from $F$ to its restriction to $A$ along $g$).

Let's unwind what that means. The value of $\overline{F}$ on anything of the form $(a, 0)$ is already determined, so the only question is what is the value of $\overline{F}$ at $b \in B$. Let $M = (A \times [1]) \amalg_{A \times \{1\}} B$. Then $\overline{F}(b)$ must be a colimit of the diagram $A \times_M M_{/b} \to A \to \mathcal{C}$. But the natural map $A \times_M M_{/b}$ is the same as $A \times_B B_{/b}$, so this is the usual notion of a pointwise Kan extesion.

Moreover: in the previous section, Lurie proves that Kan extensions along fully faithful functors exist and are unique provided the relevant colimits exist. So the Kan extension $\overline{F}$ exists if and only if the usual colimits over $A \times_B B_{/b}$ exist, by what we saw above. Whence the result.

## Shah's approach

(This is logically unnecessary for the answer to your question, but worth advertising since it's a neat trick).

The actual proof of existence in HTT, in the fully faithful case, is a little difficult. Jay Shah has a nice approach which treats the fully faithful and general case at once and is very clean. The idea is to factor $g: A \to B$ as $A \to A\times_{B^{\{1\}}}B^{[1]} \to B$. The first map is a right adjoint (with left adjoint given by projecting to $A$) and we can always Kan extend along right adjoints: just compose with the left adjoint. The second map is a cocartesian fibration, and its fibers are precisely those diagrams of interest "$A \times_B B_{/b}$". So we are reduced to the following assertion: if $\pi: E \to B$ is a cocartesian fibration, then we may left Kan extend along $\pi$ provided the colimits along the fibers exist. This is not so bad: one forms the relative overcategory for a functor $F: E \to \mathcal{C}$, i.e. a cartesian fibration $E^{(F, B)/} \to B$ whose fiber over $b$ is the overcategory for $E_b \to E \to \mathcal{C}$. The assumption is that each fiber has an initial object. But the full subcategory of fiberwise initial objects inside of a cartesian fibration is a trivial Kan fibration over its image (this is not so hard to prove directly), and so we may choose a section. This completes the proof.

## Functoriality

You ask about functoriality in the following form: fix an $\infty$-category $\mathcal{C}$ and a functor $g: A \to B$. Then (according to Riehl-Verity) $\mathcal{C}$ admits functorial pointwise left Kan extensions for $g$ if there is a left Kan extension of the evaluation map $A \times \mathcal{C}^A \to \mathcal{C}$ along $A \times \mathcal{C}^A \to B \times \mathcal{C}^A$.

By the existence result, we know that this Kan extension exists provided that, for each pair $(b, F)$, the colimit over $(A \times \mathcal{C}^A) \times_{B \times \mathcal{C}^A} (B \times \mathcal{C}^A)_{/(b, F)}$ exists. But the inclusion of $A \times_B B_{/b}$ into this category given by $(a, ga \to b) \mapsto ((a, F), ga \to B, F = F)$ is final (indeed, it has a left adjoint given by $((a, G), ga \to b, G \to F) \mapsto (a, ga \to B)$). Therefore, this colimit exists if and only if the corresponding colimit over $A \times_B B_{/b}$ exists.

In summary: if $\mathcal{C}$ admits colimits of shape $A \times_B B_{/b}$ for all $b$, then $\mathcal{C}$ admits 'functorial pointwise Kan extensions' in the sense of Riehl-Verity.

(It is also possible to deduce this functoriality formally from the existence of the left adjoint $g_!$ to the restriction $g^*$ but the above is probably simpler).