# Equivalence definition of creation of limits?

The following is at Emily Riehl’s Category theory in context, page 90.

Firstly the Definition of creation of limits:

Definition 3.3.1. : For any class of diagrams $$K : J → C$$ valued in $$C$$, a functor $$F : C → D$$ creates those limits if whenever $$FK : J → D$$ has a limit in $$D$$, there is some limit cone over $$FK$$ that can be lifted to a limit cone over $$K$$, and moreover $$F$$ reﬂects the limits in the class of diagrams.

But later the author writes the following at the same page, which I think is a weaker hypothesis than that in Definition 3.3.1, because it seems not to require the limit cone existing in $$C$$ to be a lift of a limit cone in $$D$$.

Remark 3.3.2. To ground intuition for these terms, the following metaphor may prove helpful: a functor $$F : C → D$$ maps from “upstairs” (the category $$C$$) to “downstairs” (the category $$D$$). An upstairs diagram (a functor $$K : J → C$$) sits above a downstairs diagram (by composing with $$F$$).

$$F$$ creates limits if the mere presence of a downstairs limit cone implies the existence of an upstairs limit cone, and if any cone sitting above a limit cone is a limit cone.

May I miss something? Thanks for any hint.

This is indeed a weaker hypothesis. The formal definition supposed that $$F$$ lifts and reflects limits, while the intuitive one suppose only that $$F$$ reflects limits and the existence of limits. As long as $$\mathcal C$$ has limits, the formal definition implies that $$F$$ preserves them, but the intuitive one does not, so is missing a piece of the concept.