It is a theorem on the stacks project that if a category $\mathcal{C}$ has finite products and equalizers then it has a terminal object, this theorem is at https://stacks.math.columbia.edu/tag/04AS.
But the category $\mathbb{Z}$ considered as a totally ordered set has all finite products which are given by taking the minimal element, and it vacuously have all equalizers, but has no final object. What is going wrong here?
$\mathbb{Z}$ does not, in fact, have finite limits: what's the limit of the empty diagram?
(The empty diagram counts as a finite diagram, so when we say that a category has finite limits it has to apply to the empty diagram too.)