Definition 10.2.0.6. Let $\operatorname{\mathcal{C}}$ be a category. We say that $\operatorname{\mathcal{C}}$ is *regular* if it satisfies the following conditions:

- $(1)$
The category $\operatorname{\mathcal{C}}$ admits finite limits (in particular, it admits fiber products).

- $(2)$
Every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ has an image: that is, we can write $f$ as a composition

\[ X \stackrel{f_0}{\twoheadrightarrow } Y_0 \stackrel{i}{\hookrightarrow } Y \]where $i$ is a monomorphism and $f_0$ is a regular epimorphism.

- $(3)$
The collection of regular epimorphisms is stable under the formation of pullbacks. That is, for every pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]in the category $\operatorname{\mathcal{C}}$, if $f$ is a regular epimorphism, then $f'$ is also a regular epimorphism.