Right exactness on a dense subcategory Let $F : C \to D$ be a $k$-linear functor between cocomplete $k$-linear categories, which preserves directed colimits (in particular arbitrary direct sums). Let $C' \subseteq C$ be a dense full subcategory such that the restriction $F|_{C'}$ is right exact (i.e. preserves coequalizers). Is it possible to conclude that $F$ is right exact?
If not, are there any general methods how to prove right exactness of $F$? Is there any obstruction which measures this and may be computed (like derived functors in the abelian case)? Such a situation appears in my work and I honestly have no idea how to proceed. Any comments, also about related situations, are welcome. You may also add additional assumptions on $C$ and $D$.
 A: Suppose $\mathcal{C}' \subseteq \mathcal{C}$ is dense. Then, for any category $\mathcal{J}$, there is a dense functor $\mathcal{J}^\mathrm{op} \times \mathcal{C}' \to [\mathcal{J}, \mathcal{C}]$, namely $(j, A) \mapsto \mathcal{J}(j, -) \odot A$. (This is easiest to see if you use the fact that $\mathcal{A} \to \mathcal{B}$ is dense if and only if $\mathcal{B} \to [\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is fully faithful.) Note that $\mathcal{J}^\mathrm{op} \times \mathcal{C}'$ may not embed as a subcategory of $[\mathcal{J}, \mathcal{C}]$, but this is inconsequential.
We wish to compare the functors $F \circ {\varinjlim}_\mathcal{J}$ and ${\varinjlim}_\mathcal{J} \circ [\mathcal{J}, F]$. Suppose they agree (up to isomorphism) on $\mathcal{J}^\mathrm{op} \times \mathcal{C}'$. The definition of density implies that the restriction functor $[[\mathcal{J}, \mathcal{C}], \mathcal{D}] \to [\mathcal{J}^\mathrm{op} \times \mathcal{C}', \mathcal{D}]$ is fully faithful (in particular, conservative) on the full subcategory of functors $[\mathcal{J}, \mathcal{C}] \to \mathcal{D}$ that preserve "canonical colimiting cocones", i.e. the tautological cocone from the diagram of shape $(\mathcal{J}^\mathrm{op} \times \mathcal{C}' \downarrow X)$ to $X$ for each object $X$ in $[\mathcal{J}, \mathcal{C}]$. Thus, if $F$ preserves enough colimits, then $F : \mathcal{C} \to \mathcal{D}$ preserves $\mathcal{J}$-colimits if and only if $F : \mathcal{C}' \to \mathcal{D}$ preserves $\mathcal{J}$-colimits.
The problem is to determine what exactly "enough colimits" means. If $\mathcal{J}^\mathrm{op} \times \mathcal{C}'$ has $\kappa$-small colimits and the functor $\mathcal{J}^\mathrm{op} \times \mathcal{C}' \to [\mathcal{J}, \mathcal{C}]$ preserves them, then it is enough that $F$ preserve $\kappa$-filtered colimits, but I suspect this rarely happens. 
When $\mathcal{C}$ is locally $\kappa$-presentable, $\mathcal{C}'$ is the full subcategory of $\kappa$-compact objects, and $\mathcal{J}$ is $\kappa$-small, there is an alternative argument. In that case, $[\mathcal{J}, \mathcal{C}]$ is again locally $\kappa$-presentable and the $\kappa$-compact objects in $[\mathcal{J}, \mathcal{C}]$ are precisely the diagrams of $\kappa$-compact objects in $\mathcal{C}$. (This is not completely abstract nonsense and relies on the fact that $\kappa$-filtered colimits in $\mathbf{Set}$ preserve $\kappa$-small limits!) Thus, $[\mathcal{J}, \mathcal{C}']$ is a dense subcategory of $[\mathcal{J}, \mathcal{C}]$, and $F : \mathcal{C} \to \mathcal{D}$ preserves $\mathcal{J}$-colimits if and only if $F : \mathcal{C}' \to \mathcal{D}$ preserves $\mathcal{J}$-colimits.
The situation with left exactness is somewhat better, for certain technical reasons. Given any $G : \mathcal{D} \to \mathcal{C}$ and any dense $\mathcal{C}' \subseteq \mathcal{C}$, if for all $C'$ in $\mathcal{C}'$ there exist $F C'$ in $\mathcal{D}$ and a bijection
$$\mathcal{C}(C', G D) \cong \mathcal{D}(F C', D)$$
natural in $D$, then $G : \mathcal{D} \to \mathcal{C}$ must preserve all limits, almost as if $G$ had a left adjoint. If $\mathcal{D}$ were sufficiently cocomplete, then we would even be able to construct an actual left adjoint $F : \mathcal{C} \to \mathcal{D}$.
