Isomorphism

In category theory, an isomorphism is an equivalence relation:

$$\begin{array}{rl}\text{Given }& \mathcal{C}:\mathbf{Cat}\\ & A,B:\mathcal{C}\\ & f:A\Rightarrow B\text{ is an isomorphism iff}\end{array}$$ $$\exists g:B\Rightarrow A.g\circ f=1_A\wedge f\circ g=1_B.$$