\begin{definition}[Definition:Isomorphism (Category Theory)]
Let $\mathbf C$ be a category, and let $X, Y$ be objects of $\mathbf C$.
A morphism $f: X \to Y$ is an '''isomorphism''' if there exists a morphism $g: Y \to X$ such that:
:$g \circ f = I_X$
:$f \circ g = I_Y$
where $I_X$ denotes the identity morphism on $X$.
It can be seen that this is equivalent to $g$ being both a retraction and a section of $f$.
\end{definition}