cocompletion

co- +‎ completion

cocompletion (plural cocompletions)

1. (mathematics) A full embedding whose image is a cocomplete category and for which every functor with a cocomplete image has an extension to a cocontinuous functor that is unique up to natural isomorphism.
   • 2015, Rory B. B. Lucyshyn-Wright, “Enriched algebraic theories and monads for a system of arities”, in arXiv‎:

     We show that ${\displaystyle j}$ satisfies this condition if and only if ${\displaystyle j}$ presents ${\displaystyle {\mathcal {V}}}$ as a free cocompletion of ${\displaystyle {\mathcal {J}}}$ with respect to the weights for left Kan extensions along ${\displaystyle j}$, and so we call such systems of arities eleutheric.