Let $\kappa$ be a regular cardinal. A locally small category $\mathsf{C}$ is locally $\kappa$-presentable if it is cocomplete and if it has a set of objects $S$ so that:
- Every object in $\mathsf{C}$ can be written as a colimit of a diagram valued in the subcategory spanned by the objects in $S$.
- For each object $s \in S$, the functor $\mathsf{C}(s,-) \colon \mathsf{C} \to \textup{\textsf{cat}}$ preserves $\kappa$-filtered colimits.
A functor between locally $\kappa$-presentable categories is accessible if it preserves $\kappa$-filtered colimits.