It is another question about the category $\mathcal{D}\mathcal{K}$ of all small diagrams over all small categories, its definition is here : [About the category of all small diagrams](https://mathoverflow.net/q/266597). I suppose $\mathcal{K}$ locally $\lambda$-presentable. 



> 
> Question: Take a diagram $F:I\to \mathcal{K}$ such that $I$ is
>  finitely presentable as a small category and $F$ $\lambda$-presentable
>  in the functor category $\mathcal{K}^I$. Is $F$ $\lambda$-presentable
>  in $\mathcal{D}\mathcal{K}$ ?
> 
> 
> 


The motivation of this question is that $\mathcal{D}\mathcal{K}$ has a colimit-dense category of such objects. Therefore, assuming Vopenka's principle, $\mathcal{D}\mathcal{K}$ is locally presentable. I would like to avoid the use of this principle. 



Presentable small diagrams over a locally presentable category
Maybe I have this wrong, but I think that in $\mathcal{DK}$, filtered colimits at least are constructed in the following way. Let $I$ be a $\lambda$-directed poset for simplicity, and let $G: I \to \mathcal{DK}$, $i \mapsto G\_i: D\_i \to \mathcal{K}$ be a functor, with transition maps $(D\_{ii'}: D\_i \to D\_{i'}, \gamma\_{ii'}: G\_i \Rightarrow G\_{i'} \circ D\_{ii'})$ for $i\leq i'$. Let $G\_\infty : D\_\infty \to \mathcal{K}$ denote the colimit of $G$. Then we should have $D\_\infty = \varinjlim\_i D\_i$, so an object of $\varinjlim\_{i \in I} D\_i$ is an equivalence class $[(i,d)]$ where $i \in I$ and $d \in D\_i$. And $G\_\infty$ should be given by $G\_\infty([(i,d)]) = \varinjlim\_{i \leq i'} G\_{i'}(D\_{ii'}(d))$ (where the colimit is of a diagram constructed using the $\gamma$ maps; this colimit is $\lambda$-directed).


Now suppose that $F: C \to \mathcal{K}$ is such that $C$ is $\lambda$-presentable, and $F$ takes values in the $\lambda$-presentable objects of $\mathcal{K}$. A map $F \to G\_\infty$ will consist first of a functor $f: C \to D\_\infty$; this data commutes with the colimit because $C$ is $\lambda$-presentable. Secondly, there will be a natural transformation $\mu: F \Rightarrow G\_\infty \circ f$. In components this consists of maps $\mu\_c : Fc \to G\_\infty(fc)$; the data of this component commutes with the colimit because $Fc$ is $\lambda$-presentable. The data of the whole natural transformation then commutes with the colimit because $C$ is $\lambda$-presentable; I'm not sure how to argue this conceptually, but it follows from the idea that a $\lambda$-presentable category is generated by $<\lambda$-many morphisms subject to $<\lambda$-many relations.


That is, $F$ is indeed $\lambda$-presentable, with slightly weaker hypotheses than requested.


It's interesting that standard results about presentability and fibrations like Makkai-Paré Theorem 5.3.4 don't seem to quite apply here.


