Now consider the inserter $Ins(F,G)$ between functors $F,G: \mathcal{C} \to \mathcal{D}$. Colimits are computed as in $\mathcal{C}$, and it's easy to see that the objects whose underlying $\mathcal{C}$-object is $\kappa$-presentable are $\kappa$-presentable. So it suffices to show that the colimit closure of these objects is all of $Ins(F,G)$ -- by Theorem 2.5.1 in Makkai-Paré, or by Lemma 3.7 in these notes by Mike Shulman and the usual strong-generator characterization of locally $\kappa$-presentable categories. Consider an object $F(c) \overset{\gamma}{\to} G(c)$ of $Ins(F,G)$. Then $c$ has some presentability rank $\lambda$. If $\lambda \leq \kappa$ we are done, otherwise $\lambda = \mu^+$ for some uncountable $\mu$, and $c$ is a $\mu$-sized colimit of $\kappa$-presentable objects (see Adámek and Rosický Remark 1.30). Let $K: I \to \mathcal{C}$ be such a diagram. There are two cases: either $\mu$ is regular, or $\mu$ is singular.
$\require{AMScd}
\begin{CD}
F i_\beta @>>> Fi_\alpha^0 @>>> \cdots @>>> Fi_\alpha^n @>>> Fi_\alpha @>>> Fc \\
@V{\gamma_\beta}VV @V{\gamma_\alpha^0}VV @VVV @V{\gamma_\alpha^n}VV @V{\gamma_\alpha}VV @V{\gamma}VV \\
G i_\beta @>>> Gi_\alpha^1 @>>> \cdots @>>> Gi_\alpha^{n+1} @>>> Gi_\alpha @>>> Gc
\end{CD}$$\require{AMScd}
\begin{CD}
FK i_\beta @>>> FKi_\alpha^0 @>>> \cdots @>>> FKi_\alpha^n @>>> FKi_\alpha @>>> Fc \\
@V{\gamma_\beta}VV @V{\gamma_\alpha^0}VV @VVV @V{\gamma_\alpha^n}VV @V{\gamma_\alpha}VV @V{\gamma}VV \\
GK i_\beta @>>> GKi_\alpha^1 @>>> \cdots @>>> GKi_\alpha^{n+1} @>>> GKi_\alpha @>>> Gc
\end{CD}$
In the diagram, we have set $i_\alpha = \varinjlim_{n<\omega} i_\alpha^n$ and observed that becuse this colimit is preserved by $K$, $F$, and $G$, we can define $\gamma_\alpha : Fi_\alpha \to Gi_\alpha$$\gamma_\alpha : FKi_\alpha \to GKi_\alpha$ to be the colimit of the $\gamma_\alpha^n$'s, and this is indeed how we define $i_\alpha$ and $\gamma_\alpha$. This is where we need $\kappa$ to be uncountable -- otherwise we cannot take the colimit of this chain when $\mu = \kappa$.
Choose a sequence of regular cardinals $(\mu_\alpha)$ satisfying $\kappa \leq \sum_{\beta < \alpha} \mu_\beta < \mu_\alpha < \mu$ with supremum $\mu$. Similarly to the preliminaries before, we can replace $I$ with a $\mu$-sized poset equipped with an exhaustive filtration $I_0 \subset I_1 \subset \dots \subset I$, $\cup_\alpha I_\alpha = I$ satisfying the conditions that $|I_\alpha| = \mu_\alpha$ and $I_\alpha$ is closed under $\mu_\alpha$-small colimits, which are preserved by the inclusion $I_\alpha \to I$. And we may assume that $K$ preserves $\mu$-small colimits and that $Ki$ is $\mu_\alpha$-presentable for $i \in I_\alpha$ . We construct a chain $(i_\alpha)$ with $i_\alpha \in I_\alpha$ and a natural family of morphisms $\gamma_\alpha : Fi_\alpha \to Gi_\alpha$ by taking $i_\alpha^0 = \sup (\cup_{\beta < \alpha} I_\beta)$ and performing an iterative construction as before to choose $i_\alpha$ and $\gamma_\alpha$. It is harmless to assume that $i_\alpha \in I_\alpha$ -- otherwise $i_\alpha$ first appears in $I_{\alpha'}$ for some $\alpha'>\alpha$, and we simply we simply modify our choice of $I_\beta$ for $\alpha \leq \beta < \alpha'$ by adding in $i_\alpha$ and closing under $\mu_\beta$-filtered colimits in $I_{\alpha'}$. As before, we have defined a chain in $Ins(F,G)$ whose colimit is $Fc \overset{\gamma}{\to} Gc$. Moreover for each $\alpha$, the object $Ki_\alpha$ is $\mu_\alpha$-presentable, and $\mu_\alpha$ is a regular cardinal strictly less than $\lambda$, so by the inductive hypothesis, $Fc \overset{\gamma}{\to} Gc$ is in the colimit closure of the $\kappa$-presentable objects as desired.
