zero object in nLab

Proof

Write *∈Set ** \in Set_* for the singleton pointed set. Suppose tt is terminal. Then C(x,t)=*C(x,t) = * for all xx and so in particular C(t,t)=*C(t,t) = * and hence the identity morphism on tt is the basepoint in the pointed hom-set. By the axioms of a category, every morphism f:t→xf : t \to x is equal to the composite

f:t→Idt→fx. f : t \stackrel{Id}{\to} t \stackrel{f}{\to} x \,.

By the axioms of an (Set *,∧)(Set_*, \wedge)-enriched category, since Id tId_{t} is the basepoint in C(t,t)C(t,t), also this composite is the basepoint in C(t,x)C(t,x) and is hence the zero morphism. So C(t,x)=*C(t,x) = * for all xx and therefore tt is also an initial object.

Analogously from assuming tt to be initial it follows that it is also terminal.