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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. 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.

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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.

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    $\begingroup$ I agree with the argument but I don't understand why Makkai-Paré Theorem 5.3.4 does not apply here; the pseudofunctor $\mathrm{CAT}(-,\mathcal{K}):\mathrm{Cat}^{op}\to \mathrm{CAT}$ is accessible by keeping only in $\mathrm{Cat}^{op}$ the identity natural transformations. What am I missing ? $\endgroup$ – Philippe Gaucher Jun 12 '17 at 13:39
  • $\begingroup$ Oh... I suppose you're right! I was thinking that Makkai-Paré's condition (4) for an accessible fibration seemed to be violated, but now I think I was just confused! $\endgroup$ – Tim Campion Jun 12 '17 at 20:17

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