# Question on the definition of a locally presentable category

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)?

• Thank you, Zhen Lin. But when Vopěnka's principle is used, the small category $K$ changes, doesn't it? – user8463524 May 7 '15 at 6:33
• You do not need to change $K$. The inclusion is automatically accessible in that case. – Zhen Lin May 14 '15 at 9:51
• 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. – Zhen Lin May 14 '15 at 12:51