Definition:Isomorphism (Category Theory)

Definition
Let $\mathcal C$ be a category, and let $X, Y$ be objects of $\mathcal C$.

A morphism $f: X \to Y$ is an isomorphism if there exists a morphism $g: Y \to X$ such that:
 * $g \circ f = \operatorname{id}_X$
 * $f \circ g = \operatorname{id}_Y$

where $\operatorname{id}_Z$ denotes the identity morphism on an object $Z$ of $\mathcal C$.

Linguistic Note
The word isomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.