In the linked question here, the user demonstrates two examples of extension of morphisms using Zorn's Lemma arguments, and I've seen the same pattern to extend morphisms before in other sources.
However, none of them contain a verification of the condition of Zorn's lemma that every totally ordered subset has an upper bound. They jump straight from establishing the order relation to the existence of a maximal element.
Presumably this is because the verification follows a routine pattern that is obvious if you have seen it before. But supposing I haven't, how do we know that these posets meet the upper bound condition of Zorn's Lemma?