Implication Details

Claim: If a category has pullbacks, then it has coreflexive equalizers.

Proof: Assume that $f,g : A \rightrightarrows B$ is a coreflexive pair. Choose a morphism $r : B \to A$ with $r \circ f = r \circ g = \id_A$. By assumption, $f,g$ have a pullback $$\begin{CD} P @>{u}>> A \\ @V{v}VV @VV{g}V \\ A @>>{f}> B \end{CD}$$ Then $$u = r \circ f \circ u = r \circ g \circ v = v.$$ Thus, $u$ equalizes $f$ and $g$. If $w : T \to A$ is another morphism equalizing $f$ and $g$, we have a commutative diagram $$\begin{CD} T @>{w}>> A \\ @V{w}VV @VV{g}V \\ A @>>{f}> B, \end{CD}$$ so that, by the universal property of the pullback, there is a unique morphism $h : T \to P$ with $u \circ h = w$ and $v \circ h = w$. The second equation is redundant because $u = v$. Thus, we have shown that $w$ factors uniquely through $u$, showing that $u$ is an equalizer of $f$ and $g$.

This implication has a dual.