Definition 1.1.2 (Isomorphism). Let $\catc$ be a category, $A, B \in \obj{\catc}$, and $f \in \mor{A, B}$, then $f$ is an isomorphism if there exists $g \in \mor{B, A}$ such that $g \circ f = \text{Id}_{A}$ and $f \circ g = \text{Id}_{B}$.

For any $A, B \in \obj{\catc}$, $A$ and $B$ are isomorphic if there exists an isomorphism $f \in \mor{A, B}$.