nLab
subterminal object

An object $U$ in a category $C$ is subterminal if any two morphisms with target $U$ are equal. In other words, $U$ is subterminal if for any object $X$, there is at most one morphism $X\to U$.

If $C$ has a terminal object $1$, then this is equivalent to saying that the unique map $U\to 1$ is monic; hence the name “sub-terminal.” If the product $U\times U$ exists, it is equivalent to saying that the diagonal $U\to U\times U$ is an isomorphism.

The subterminal objects in a topos can be viewed as its “external truth values.” For example, in the topos $\mathrm{Sh}(X)$ of sheaves on a topological space $X$, the subterminal objects are precisely the open sets in $X$.