Definition of Grothendieck topology

Question

I'm learning about Grothendieck topology in https://ncatlab.org/nlab/show/Grothendieck+topology
There are two definitions that are Defintion2.1 and Definition 3.1 in the page.
I think those are equivalent, but I can't prove.

Defintion2.1 and 3.1 are equivalent??
If not , please tell me counterexamples.

Answer 1

They are not equivalent. Definition 3.1 is more general because it does not require the category to have any pullbacks.

However, if the underlying category does have pullbacks, then the two definitions do become equivalent. To pass from Definition 2.1 to Definition 3.1, assign to every covering family $ϕ$ of Definition 2.1 the corresponding sieve, given by the intersection of all sieves that contain $ϕ$. The resulting collection of sieves is a Grothendieck topology in the sense of Definition 3.1.