An example of a Fra\"iss\'e class without a Kat\v{e}tov functor

We disprove a conjecture from [W. Kubi\'s, D. Ma\v{s}ulovi\'c, Kat\v{e}tov functors, preprint, http://arxiv.org/abs/1412.1850] by showing the existence of a Fra\"iss\'e class $\mathcal{C}$ which does not admit a Kat\v{e}tov functor. On the other hand, we show that the automorphism group of the Fra\"iss\'e limit of $\mathcal{C}$ is universal, as it happens in the presence of a Kat\v{e}tov functor.