  In this paper we characterize those accessible $\mathcal V$-categories that
have limits of a specified class. We do this by introducing the notion of
companion $\mathfrak C$ for a class of weights $\Psi$, as a collection of
special types of colimit diagrams that are compatible with $\Psi$. We then
characterize the accessible $\mathcal V$-categories with $\Psi$-limits as those
accessibly embedded and $\mathfrak C$-virtually reflective in a presheaf
$\mathcal V$-category, and as the $\mathcal V$-categories of $\mathfrak
C$-models of sketches. This allows us to recover the standard theorems for
locally presentable, locally multipresentable, and locally polypresentable
categories as instances of the same general framework. In addition, our theorem
covers the case of any weakly sound class $\Psi$, and provides a new
perspective on the case of weakly locally presentable categories.
