Isomorphism (In a Category)

Let $\mathscr{C}$ be a category , and let $a$ and $b$ be object in $\mathscr{C}$. We call a morphism $f:a \rightarrow b$ an isomorphism between $a$ and $b$ provided there exists a morphism $g:b \rightarrow a$ such that:

If such an isomorphism exists between $a$ and $b$, we call $a$ and $b$ isomorphic.