Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors

Let $F:\mathcal{C}\to \mathcal{D}$ be a functor (Covariant functor, identity functor, composite functor, and contravariant functor) and fix an indexing category $\mathcal{J}$.

• $F$ preserves $\mathcal{J}$-limits if the image under $F$ of every limiting cone over $D:\mathcal{J}\to \mathcal{C}$ is limiting over $FD$.
• $F$ reflects $\mathcal{J}$-limits if a cone over $D$ is limiting whenever its image is limiting.
• In the ordinary isomorphism-invariant sense used here, $F$ creates $\mathcal{J}$-limits if every limiting cone over $FD$ is isomorphic as a cone (Natural isomorphism) to the image of a cone over $D$, that source cone is limiting, and any source cone whose image is limiting is limiting. This does not require an on-the-nose lift of the target apex.
• The functor strictly creates $\mathcal{J}$-limits if every limiting cone over $FD$ has a unique lift with exactly the same apex and legs under $F$, and that lifted cone is limiting. Strict creation therefore contains data not demanded by ordinary isomorphism-invariant creation.

The terms preserves, reflects, and creates colimits use cocones and colimits (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties). A functor is continuous if it preserves all small limits and cocontinuous if it preserves all small colimits.