Definition 1.2.2 (Isomorphism).label Let $\frak{C}$ be a category and $X,Y\in \ob{\frak{C}}$ then $X$ and $Y$ are isomorphic, denoted by $X\iso Y$, if there are morphisms $f\in \mor{}{X }{Y },g\in \mor{ }{Y }{X }$ such that $f\circ g=\text{Id}_{Y}$ and $g\circ f=\text{Id}_{X}$.