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According to nlab, a category $C$ is called locally presentable if it is accessible and has all small colimits.

Moreover, one can show, that this conditions are equivalent to the condition of $C$ being a reflective subcategory of some presheaf-category $PSh(K)$ for a small category $K$ such that the inclusion $i\colon C\hookrightarrow PSh(K)$ is an accessible functor.

I once saw the definition of $C$ being locally presentable if $C$ is just a reflective subcategory of some presheaf-category $PSh(L)$?

Are this conditions equivalent? If yes, do I have to change the category $K$ in order to get the last 'definition' (i.e. is the second definition just redundant and the inclusion is automatically accessible)?

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There is absolutely no reason to expect to be able to get accessibility for free. On the other hand, I know of no explicit counterexamples – and indeed, if you assume Vopěnka's principle, then you can show that every reflective subcategory of a locally presentable category is automatically a locally presentable category.

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  • $\begingroup$ Thank you, Zhen Lin. But when Vopěnka's principle is used, the small category $K$ changes, doesn't it? $\endgroup$ – user8463524 May 7 '15 at 6:33
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    $\begingroup$ No, it does not. $\endgroup$ – Zhen Lin May 7 '15 at 6:58
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    $\begingroup$ They are generators but they are not necessarily presentable. $\endgroup$ – Zhen Lin May 14 '15 at 8:09
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    $\begingroup$ You do not need to change $K$. The inclusion is automatically accessible in that case. $\endgroup$ – Zhen Lin May 14 '15 at 9:51
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    $\begingroup$ Given a fully faithful functor $\mathcal{C} \to \mathbf{Psh} (\mathcal{K})$ with a left adjoint, if $\mathcal{C}$ is accessible, then the functor is also accessible. This is an easy exercise. $\endgroup$ – Zhen Lin May 14 '15 at 12:51

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