In Category Theory in Context, Riehl defines creating limits as follows, for a functor $F: \mathbb{C} \to \mathbb{D}$ and diagram $D: I \to \mathbb{C}$:
$F$ creates limits if whenever $FD: I \to \mathbb{D}$ has a limit in $\mathbb{D}$, there is some limit cone over $FD$ that can be lifted to a limit cone over $D$, and moreover $F$ reflects limits.
In other words, using terminology similar to this answer, $F$ creates limits iff:
- it lifts limits up to isomorphism (in particular, it detects limits);
- it reflects limits.
Then the author says:
With the exception of the equivalences the functors that one meets in practice that create certain limits and colimits tend to do so strictly in the sense of the following definition.
Definition 3.3.7. A functor $F: \mathbb{C} \to \mathbb{D}$ strictly creates limits if for any diagram $D : I \to \mathbb{C}$ and any limit cone over $FD$,
- there exists a unique lift of that cone to a cone over $D$, and
- moreover, this lift defines a limit cone in $\mathbb{C}$.
As pointed out, I can think of an equivalence of categories that creates limits but not strictly, for example the equivalence $\mathrm{FinSet} \to \mathbb{N}$ onto its skeleton. But I can't think of any other type of example. What is an example of a functor that creates limits but not strictly, other than an equivalence of categories?