Extending functors defined on dense subcategories This question will fade from very specific to very generic.
Consider an $\lambda$-accessible category $\mathcal{K}$ and let's call $\text{Pres}_{\lambda} \mathcal{K}$ its $\lambda$-presentables.
Is the following statement true?

Given a category with directed colimits $ \mathcal{C}$ and a functor $F:  \text{Pres}_{\lambda} \mathcal{K}\to \mathcal{C}$, there is an extension $\bar{F} : \mathcal{K} \to \mathcal{C}.$

I do have an idea to prove it.
Consider objects $K, K' \in \mathcal{K}$ and a map $K \stackrel{f}{\to} K'$. I need to give a definition for $\bar{F}(K), \bar{F}(K'), \bar{F}(f)$.
This is my attempt: Choose a directed diagram of $\lambda$-presentables $K_i$ such that $K = \text{colim }K_i$ and same for a diagram $K'_j$ whose colimit will be $K'$.
The map $K_i \to K \to K'$ factors through $K_i \to K \to K'_{\bar{i}}$ because of presentability of $K_i$. We call this map $f_i$.
Moreover there is a natural map $$\text{colim}F(K_{\bar i}) \stackrel{i}{\to} \text{colim}F(K_{ i})$$ 
I pose $$\bar{F}(K) = \text{colim}F(K_i) $$ $$\bar{F}(f) =  i \circ \text{colim}(F(f_i)).$$ 
Probably with this definition composition will not work, but maybe one can make it work.

Now question gets more vague. Consider a category $\mathcal{K}$ and  $\mathcal{G}$ the subcategory generated by a strong generator.
Can the following statement be true with some additional hypotesis?

Given a category with directed colimits $ \mathcal{C}$ and a functor $F:  \mathcal{G}\to \mathcal{C}$, there is an extension $\bar{F} : \mathcal{K} \to \mathcal{C}.$

 A: $\mathcal K$ is a free cocompletion of its $\lambda$-presentables under $\lambda$-filtered colimits. Thus $\bar{F}$ exists and preserves $\lambda$-filtered colimits.
A: $\require{AMScd}
\require{graphicx}
\def\K{\mathcal{K}}
\def\Pres{\text{Pres}}
\def\C{\mathcal C}
\def\Lan{\text{Lan}}
$
You would like to define $\bar F$ as a suitable Kan extension: in the diagram
$$
\begin{CD}
\Pres_\lambda \K @>F>> \C\\
@ViVV @.\\
\K
\end{CD}
$$
(where $i$ is the natural embedding of $\lambda$-presentable objects into $\K$) it is natural to define $\bar F$ as $\Lan_iF$, so that
$$
\bar F (K) \cong \varinjlim_{(A,f)\in (i\downarrow K)} FA
$$
since of course the comma category $(i\downarrow K)$ is the category of elements of the functor $\hom(i\,\_\, , K)$, a sufficient condition for this colimit to exist in $\C$ is that this latter category is filtered.
But this is well-studied (actually, in a properly dualized form)! For a functor $G : \mathcal{A}\to Set$ the following conditions are equivalent:


*

*$G$ commutes with finite limits;

*The Yoneda extension $\Lan_YG$ commutes with finite limits;

*the category $\text{Elts}(G)°$ is filtered.


(this is Borceux,1 6.1.3 by the way)
