Initial Object is Unique

Theorem

Let $\mathbf C$ be a metacategory.

Let $0$ and $0'$ be two initial objects of $\mathbf C$.

Then there is a unique isomorphism $u: 0 \to 0'$.

Hence, initial objects are unique up to unique isomorphism.

Proof

Consider the following commutative diagram:

$\quad\quad \begin{xy} <-4em,0em>*+{0} = "M", <0em,0em> *+{0'}= "N", <0em,-4em>*+{0} = "M2", <4em,-4em>*+{0'}= "N2", "M";"N" **@{-} ?>*@{>} ?*!/_.6em/{u}, "M";"M2" **@{-} ?>*@{>} ?*!/^.6em/{\operatorname{id}_0}, "N";"M2" **@{-} ?>*@{>} ?*!/_.6em/{v}, "N";"N2" **@{-} ?>*@{>} ?*!/_1em/{\operatorname{id}_{0'}}, "M2";"N2"**@{-} ?>*@{>} ?*!/^.6em/{u}, \end{xy}$

It commutes as each of the morphisms in it originates from an initial object, and hence is unique.

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

$\blacksquare$

Also see

• Terminal Object is Unique

Sources

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