A functor$F\colon \mathcal{C}\to \mathcal{D}$ is said to liftlimits of a particular shape $I$ if for any diagram $J:I\to C$, any limiting cone for $F \circ J$ in $\mathcal{D}$ is the image of a limiting cone for $J$ in $\mathcal{C}$.

Terminological remarks

Lifting limits is closely related to creating them. The relationships between these notions were the subject of a post by Aleks Kissinger at the categories mailing list, here, but there is some dispute about its correctness.