${\rm Fun}^R(C_1^\text{op},C_2)$ is a presentable category I am stuck in proving Lemma 5.25 in Moritz Groth's notes: I am asked to prove that for any two presentable categories $C_1,C_2$ the category of limit preserving functors ${\rm Fun}^R(C_1^\text{op},C_2)$ is again presentable.
Groth solves the particular case $C_1=[E_1^{op},{\bf Sets}]$, appealing thm 2.39 in Adamek-Rosicky's book; then he says "With some more effort the general case can also be established."
The proof given by Groth in that particular case seems to rely on the equivalence $Adj(\widehat{E_1},C_2^{op})\cong Fun(E_1,C_2^{op})$ established via "nerve-realization", so I'm not able to generalize it.
What is the idea behind the general case?
 A: *

*A limit-preserving functor $\mathcal{C}_1^\mathrm{op} \to \mathcal{C}_2$ is a colimit-preserving functor $\mathcal{C}_1 \to \mathcal{C}_2^\mathrm{op}$.

*If $\mathcal{C}_1$ is a locally $\kappa$-presentable category and $\mathcal{A}_1$ is the full subcategory of $\kappa$-compact objects in $\mathcal{C}_1$, then the category of $\kappa$-accessible functors $\mathcal{C}_1 \to \mathcal{C}_2^\mathrm{op}$ is equivalent to the functor category $[\mathcal{A}_1, \mathcal{C}_2^\mathrm{op}]$.

*The full subcategory of colimit-preserving functors $\mathcal{C}_1 \to \mathcal{C}_2^\mathrm{op}$ can be identified with the full subcategory of $\kappa$-small colimit-preserving functors $[\mathcal{A}_1, \mathcal{C}_2^\mathrm{op}]$: see here. Thus the full subcategory of limit-preserving functors $\mathcal{C}_1^\mathrm{op} \to \mathcal{C}_2$ can be identified with the full subcategory of $\kappa$-small limit-preserving functors $\mathcal{A}_1^\mathrm{op} \to \mathcal{C}_2$.

*$\mathcal{A}_1$ has all $\kappa$-small colimits, because $\mathcal{C}_1$ is cocomplete. Thus the category of $\kappa$-small limit-preserving functors $\mathcal{A}_1^\mathrm{op} \to \mathcal{C}_2$ is the category of models for a limit sketch in the locally presentable category $\mathcal{C}_2$, which is again locally presentable by Theorem 2.60 in [Adámek and Rosický]. (It is clear that any such category is complete, and it is known that a complete accessible category is locally presentable.)
