Uniqueness of Initial Object, or why must a morphism from an object to itself be the identity?

Question

The definition of an initial object in a category $\mathscr{C}$ is defined as an object that only has one map going to each object in $\mathscr{C}$. A basic result supposedly says that any two initial objects in $\mathscr{C}$ are isomorphism.

The proof of this, I would imagine, goes something along the lines of, let $A,B\in\mathscr{C}$, then we have a unique map $f_{AB}:A\to B$ and $f_{BA}:B\to A$; since there $f_{AA}:A\to A$ and $f_{BB}:B\to B$ are unique, we get that $f_{AB}\circ f_{BA}=f_{AA}$ and $f_{BA}\circ f_{AB}=f_{BB}$. If $f_{AA}=\operatorname{Id}_A$ and $f_{BB}=\operatorname{Id}_B$ then we would be done, very good.

However, I fail to see why this should hold. After all, the uniquness of $f_{AA}$ and $f_{BB}$ only says that they must be idempotent, i.e. $f_{AA}\circ f_{AA}=f_{AA}$, and similarly for $f_{BB}$. Obviously the identity map satisfies idempotency, but so do many other maps, for example a constant map (of course for objects with structure, eg. a group or a ring there is only one constant map). So in my head at least I can very easily imagine such a category in which this is the case.

Please explain why I am wrong.

Answer 1

Because the identity arrow always exists from the definition of a category, the morphism from an initial object to itself must be the identity.

Answer 2

Let $I$ be an initial object in a category $\mathscr C$. Then for every $\mathscr C$-object $X$ there exists a unique morphism $I\xrightarrow{f_X}X$. In particular, there is a unique morphism $I\xrightarrow{f_I}I$. But $\mathscr C$ has identities by definition, so $I\xrightarrow{\DeclareMathOperator{id}{id}\id_I}I$ is another morphism in $\mathscr C$. It follows that $f_I=\id_I$ by the uniqueness of $f_I$.