_(Third time's the charm? -- old wrong answer left deleted because its comments don't apply here)_
Consider the following category with objects $\\{1,2,3,4,5\\}$.
1 --s--> 2 --f--> 3 --g--> 4 --p--> 5
-------h------> --q-->
There's an arrow $n\to m$ whenever $n\le m$, and these arrows are unique except for the following cases:
* $fs \
e h: 1\to 3$
* $p \
e q: 4\to5$
* $pg \
e qg : 3 \to 5$
Then $gf$ is an equalizer of $p$ and $q$, as seen by inspecting all arrows that end at $4$:
* $g$ and $\mathrm{id}_4$ are out because $pg\
e qg$.
* $gf$ is the equalizer itself.
* $gfs$ satisfies $p(gfs)=q(gfs)$. It factors through $gf$ as it should, and the mediating arrow $s$ is trivially unique.
* $gh$ is the same arrow as $gfs$.
However, $f$ is not an equalizer. In particular it is not an equalizer of $pg$ and $qg$ because $(pg)h=(qg)h$ yet $h$ does not factor through $f$.