STATEMENT: Let $\mathcal{C}$ be a category. An object $P$ of $\mathcal{C}$ is called universally attracting if there exists a unique morphism of each object of $\mathcal{C}$ into $P$, and is called universally repelling if for each object of $\mathcal{C}$ there exists a unique morphism of $P$ into this object.
When the context makes our meaning clear, we shall call objects $P$ as above universal. Since a universal oobject P admits the identity morphism into itself, it is clear that if $P,P'$ are two universal objects in $\mathcal{C}$, then there exists a unique isomorphism between them.
QUESTION:Why does there exist an isomorphism between $P$ and $P'$? Also when Lang says "into" something, does he mean an injective map?
Let $P,P'$ be two universally repelling objects. Then there is a unique $f:P\to P'$, and a unique $g:P'\to P$. The composition $g\circ f$ is a morphism $P\to P$, and by universality we conclude $g\circ f=id_P$. For the same reason also $f\circ g=id_{P'}$, which makes each of $f,g$ an isomorphism.
A similar argument proves the claim for two universally attracting objects.