FALSE: pullbacks preserve epimorphisms in every category with pullbacks

Let $i:{\mathbb{Q}}_{\mathbb{R}}\hookrightarrow \mathbb{R}$ be the inclusion of the rationals into the real line, regarded as Hausdorff spaces. By [L3], ${\mathbb{Q}}_{\mathbb{R}}$ is dense in $\mathbb{R}$, so [L2] says that $i$ is epic: any two continuous maps out of $\mathbb{R}$ into a Hausdorff space that agree on ${\mathbb{Q}}_{\mathbb{R}}$ are equal.

Choose an irrational point $x\in \mathbb{R}$ using [L3], and let $j:\{x\}\hookrightarrow \mathbb{R}$ be the inclusion. The pullback of $i$ along $j$ is the map $\mathrm{\varnothing }\to \{x\}$, because $\{x\}\cap {\mathbb{Q}}_{\mathbb{R}}=\mathrm{\varnothing }$. Let $D=\{0,1\}$ with the discrete topology, which is Hausdorff by [L1]. The two constant maps $\{x\}\to D$ are distinct, but their composites with $\mathrm{\varnothing }\to \{x\}$ are equal. So the pullback map is not epic, and the universal statement is false.