Terminal objects are isomorphic

Suppose \(Z,Z'\) are both terminal objects. Therefore there exist unique maps \(Z \overset{a}{\underset{b}{\rightleftarrows}}Z'\)

Composing these we get \(Z \xrightarrow{a;b} Z\), but this is forced to be the identity map because there is only one morphism from \(Z\) to itself and we have to have an identity.

Therefore we can talk about ’the terminal object’ as if there were only one.