Counterexample of pullback lemma. I'm looking for a counterexample of the pullback lemma, i.e., a diagram
$$
\require{AMScd}
\begin{CD}
A @>{}>> B @>>> C\\
@VVV @VVV @VVV\\
D @>>> E @>>> F
\end{CD}
$$
such that left square and outer square are pullbacks but right square is not a pullback. I have tried hard but I can't find a counterexample. 
Any hint would be appreciated.

The paper The other pullback lemma by Michal Przybylek exactly deals with this kind of situation. The introduction mentions two counterexamples. See also this mathoverflow thread. There you can also find a very formal counterexample by Eric Wofsey.
