In any category with finite limits, the kernel pair of a morphism $f \colon X \to Y$ is the pullback of $f$ along itself:
\(\xymatrix{ R \ar[d]_s \ar[r]^t \ar@{}[dr]\mid(.2){\displaystyle\lrcorner} & X \ar[d]^f \\ X \ar[r]_f & Y}\) These maps define a monomorphism $(s,t) \colon R \rightarrowtail X \times X$, so the object $R$ is always a subobject of the product $X \times X$. In $\textup{\textsf{cat}}$, a subset $R \subset X \times X$ defines a relation on $X$. Indeed, subobjects defined by kernel pairs are always equivalence relations, in the following categorical sense:
- There is a reflexivity map $\rho$ defined by \(\xymatrix{ X \ar@{-->}[dr]^{\rho} \ar@/^/[drr]^{1_X} \ar@/_/[ddr]_{1_X} \\ & R \ar[d]_s \ar[r]^t \ar@{}[dr]\mid(.2){\displaystyle\lrcorner} & X \ar[d]^f \\ & X \ar[r]_f & Y}\)
that is a section of both $s$ and $t$, i.e., that defines a factorization of the diagonal \(\xymatrix@R=10pt{ X \ar[rr]^{(1_X,1_X)} \ar[dr]_{\rho} & & X \times X \\ & R \ar@{ >->}[ur]_{(s,t)}}\)
- There is a symmetry map $\sigma$ defined by \(\xymatrix{ R \ar@{-->}[dr]^{\sigma} \ar@/^/[drr]^{s} \ar@/_/[ddr]_{t} \\ & R \ar[d]_s \ar[r]^t \ar@{}[dr]\mid(.2){\displaystyle\lrcorner} & X \ar[d]^f \\ & X \ar[r]_f & Y}\) so that $t \sigma =s$ and $s \sigma = t$.
- There is a transitivity map $\tau$ whose domain is the pullback of $t$ along $s$
\(\xymatrix{ R \times_X R \ar[d]_{\tilde{s}} \ar[r]^-{\tilde{t}} \ar@{}[dr]\mid(.2){\displaystyle\lrcorner} & R \ar[r]^t \ar[d]_s & X \\ R \ar[r]^t \ar[d]_s & X \\ X}\) This diagram defines a cone over the pullback defining $R$ and thus induces a map
\(\xymatrix{ R\times_X R\ar@{-->}[dr]^{\tau} \ar@/^/[drr]^{t\tilde{t}} \ar@/_/[ddr]_{s\tilde{s}} \\ & R \ar[d]_s \ar[r]^t \ar@{}[dr]\mid(.2){\displaystyle\lrcorner} & X \ar[d]^f \\ & X \ar[r]_f & Y}\) so that $s\tau = s \tilde{s}$ and $t \tau = t\tilde{t}$.
An equivalence relation in a category $\mathsf{C}$ with finite limits is a subobject $(s,t) \colon R \rightarrowtail X \times X$ equipped with maps $\rho$, $\sigma$, and $\tau$ commuting with the morphisms $s$ and $t$ in the manner displayed in the diagrams of \eqref{itm:ker-pair-i}, \eqref{itm:ker-pair-ii}, and \eqref{itm:ker-pair-iii}. When it exists, the coequalizer of the maps $s,t \colon R \rightrightarrows X$ of an equivalence relation defines a quotient object $e \colon X \twoheadrightarrow X_{/R}$. In $\textup{\textsf{cat}}$, $X_{/R}$ is the set of $R$-equivalence classes of elements of $X$. For equivalence relations arising as kernel pairs, there is a unique factorization
\(\xymatrix{ X \ar[rr]^f \ar@{->>}[dr]_e & & Y \\ & X_{/R} \ar@{-->}[ur]_m}\) In good situations, such as when $\mathsf{C}$ is a \emph{Grothendieck topos} (see \S\ref{sec:topos}), the map $m$ is a monomorphism and this factorization defines the image factorization of the map $f$: the monomorphism $X_{/R} \rightarrowtail Y$ identifies the image $X_{/R}$ as a subobject of $Y$.