Terminal object ($1$)

A terminal object receives a unique morphism from every object.

A weakly terminal set is a set of objects $(W_i)$ of a category such that every object admits at least one morphism into some $W_i$. It is the dual of a weakly initial set.

An object $S$ is subterminal when its unique morphism $S\to 1$ to the terminal object is a monomorphism. Equivalently, every object has at most one morphism to $S$. Subterminal objects form the preorder $\mathrm{Sub}(1)$.

In a sheaf topos on a topological space, subterminal sheaves correspond to open sets $U$. Their sections over $V$ are a singleton if $V\subseteq U$ and empty otherwise. An open cover gives a jointly epimorphic family of these subobjects. The open cover criterion for a local sheaf topos follows by applying a colimit-preserving global-sections functor to that family.