Talk:Initial and terminal objects

Shall we not avoid to talk about "Category of semi-groups" or "Category of non-empty Sets"
The formal definition of a category implies the existence of an identity arrow for each object. Because semi-groups do not have an identity element, I guess we should not call them "category of semigroups". Shall we not call them "questionable category of semigroups" or something in that spirit? — Preceding unsigned comment added by 178.197.234.69 (talk) 14:12, 24 October 2016 (UTC)
 * In the case of categories whose objects are sets or which have an underlying set, the identity arrow is the identity mapping from the object to itself. This is the case here. For example, in the category of non-empty sets, the objects are sets and the arrows are mappings from a set to another (or to the same) set. This has to not be confused with the category that can be associated to a specific monoid, which has only one object and whose arrows are the elements of the monoid. Contrarily to preceding examples the category associated to a monoid has only one identity element, while the category of sets (or of monoids) has many identity arrows (one for each set or monoid). D.Lazard (talk) 16:28, 24 October 2016 (UTC)

Figure is wrong?
The figure seems to show an object {0} that is terminal, but not initial (no arrows from {0} to any of the other elements of A). — Preceding unsigned comment added by 98.207.232.114 (talk) 22:25, 18 February 2018 (UTC)