Unital Category

A Category $\mathcal{C}$ is unital if:

• $\mathcal{C}$ has a Zero Object $0$.
• $\mathcal{C}$ has Finite Limits
• For a Finite family $I\to X_{i}$, the family of maps $\langle\lambda j\;.\sigma_{i}(j)\rangle:X_{i}\to\prod_{i:I}X_{i}$ defined via

  $\par\sigma_{i}(j)=\begin{cases}mathrm{id}{ifx..lse\par i}&i=j\\ 0&i\neq j\\ \end{cases}\par$

  is a Strongly Epimorphic Family.

References

• Mal’cev, Protomodular, Homological and Semi-Abelian Categories, Section 1.2