Isomorphism

In category theory, a morphism $f:\mathcal{C}[a,b]$ is an isomorphism iff there is an inverse morphism $g:\mathcal{C}[b,a]$, such that:

$$\begin{array}{r}f\circ g=1_B\\ g\circ f=1_A\end{array}$$

Remarks

• $f:A\to B$ is an isomorphism in Set iff $f$ is a bijection i.e. $f$ is a injection (monic) and surjection (epic), however it is not the case in all categories that monic $\wedge$ epic $⟹$ iso with the ordinal category $2$ being a counter-example.