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:


 * $\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.

Also see

 * Initial Object is Unique