Definition 1.2.3 (Universal Objects).label An initial (resp. final) object in a category $\frak{C}$ is an object $A\in \ob{\frak{C}}$ such that for each $X\in \ob{\frak{C}}$ the set $\mor{}{A }{X }$ (resp. $\mor{}{X }{A }$) has a single element. Initial and Final objects if exists in a category is unique up to unique isomorphism.
An extremal object is an object that is initial and/or final. A zero object is an object that is initial and final.
Proof. Let $A,A'$ be initial objects of a category $\frak{C}$ then there are unique morphisms $f\in \mor{}{A }{A'},g\in \mor{}{A' }{A }$ so $g\circ f\in \mor{}{A }{A }$ hence $g\circ f=\text{Id}_{A}$ and similarly $f\circ g=\text{Id}_{A}'$ showing $A\iso A'$. Since $\mor{}{A }{A'}$ has a single element, this isomorphism is unique. By Definition 1.2.4 we have also shown the property for final objects.$\square$
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