* See ref. $[288]$ in the bibliography for category theory and algebraic topology.

Examples:

1. Category of cross modules of groups, and the category of categorical groups
2. Category $\bf{modB}$ of finite-dimensional right-B modules, and the category $\mathcal{C} / \Sigma T$ of ideals of morphisms of a category $\mathcal{C}$ which factor through a direct some of ideal copies $\Sigma T$ .
3. The category of crossed modules of $R$ -algebroids is equivalent to the category of double $R$ -algebroids with thin structure (Brown and Mosa, 1986, 2008.)
4. The categories of crossed modules of algebroids and of double algebroids with a connection pair are equivalent.

Bibliography

1

Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.