Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition/Proof 2

Proof
Let the mapping $f: \Z_4 \to U_\C$ be defined as:

From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $f$ is an isomorphism:

Cayley Table of Integers Modulo $4$
The Cayley table for $\struct {\Z_4, +_4}$ is as follows:

Cayley Table of Gaussian Integer Units
The Cayley table for $\struct {U_\C, \times}$ is as follows: