Terminal Object is Unique

Theorem

Let $\mathbf C$ be a metacategory.

Let $1$ and $1'$ be two terminal objects of $\mathbf C$.

Then there is a unique isomorphism $u: 1 \to 1'$.

Hence, terminal objects are unique up to unique isomorphism.

Proof

Consider the following commutative diagram:

$\quad\quad \begin{xy} <-4em,0em>*+{1} = "M", <0em,0em> *+{1'}= "N", <0em,-4em>*+{1} = "M2", <4em,-4em>*+{1'}= "N2", "M";"N" **@{-} ?>*@{>} ?*!/_.6em/{u}, "M";"M2" **@{-} ?>*@{>} ?*!/^.6em/{\operatorname{id}_1}, "N";"M2" **@{-} ?>*@{>} ?*!/_.6em/{v}, "N";"N2" **@{-} ?>*@{>} ?*!/_1em/{\operatorname{id}_{1'}}, "M2";"N2"**@{-} ?>*@{>} ?*!/^.6em/{u}, \end{xy}$

It commutes, as each of the morphisms in it points to a terminal object, and hence is unique.

Thus, $v$ is an inverse to $u$, and so $u$ is an isomorphism.

$\blacksquare$

Also see

• Initial Object is Unique

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Proposition $2.10$