# Generalization of terminal object

Is it useful to consider this slight generalization of the terminal object? Within a given category, an object $A$ is called a pseudo-terminal object if for any object $X$ there exists at most one morphism from $X$ to $A$.

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Notice that if we adjoin a terminal object to the category then any of your "pseudo-terminal" objects would become subobjects of the new terminal object. This suggests that calling them "subterminal objects" may not be completely off-the-mark. –  Zhen Lin Nov 3 '12 at 10:36

It is worth noting though that the notion of a weak terminal object, that is an object $T$ such that for every object $X$ there is an arrow $X\to T$, is sometimes used. In general, weakening 'unique existence' to 'existence, unique up to something' is an important general line of investigation (see, e.g., weakening a topological group to the notion of $A_\infty$ -space). The notion you offer above seems to go in a different direction: Weaken a notion from 'unique existence' to 'unique existence, or maybe no existence at all'. This is somewhat related to the difference between functions and relations (depending on what you have in mind to do with such objects, you might want to have a look at the category $Rel$ of sets and relations).