Definition 1.2.1 (Universal Object). Let $\catc$ be a category and $P \in \obj{\catc}$, then $P$ is...
universally attracting if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{A, P}$.
universally repelling if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{P, A}$.
If $P$ is universally attracting or repelling, then $P$ is a universal object.
If $P, Q \in \obj{\catc}$ are both universally attracting/repelling, then they are isomorphic.
Proof. By assumption, there exists morphisms $f \in \mor{P, Q}$ and $g \in \mor{Q, P}$. Since $f \circ g \in \mor{Q, Q}$ and $g \circ f \in \mor{P, P}$ are unique, $f \circ g = \text{Id}_{Q}$ and $g \circ f = \text{Id}_{P}$. Thus $f$ is an isomorphism.$\square$