An object in a category is subterminal if any two morphisms with target are equal. In other words, is subterminal if for any object , there is at most one morphism .
If has a terminal object , then this is equivalent to saying that the unique map is monic; hence the name “sub-terminal.” If the product exists, it is equivalent to saying that the diagonal is an isomorphism.
The subterminal objects in a topos can be viewed as its “external truth values.” For example, in the topos of sheaves on a topological space , the subterminal objects are precisely the open sets in .