In Lurie's Kerodon it is mentioned as a remark that none of the standard $n$-simplices, i.e. the representable simplicial sets, are Kan complexes. I am curious to see a proof of this 'straight from the definition'. That is, I would like to see why we can't extend every morphism $\Lambda^k_i\to \Delta^n$ to a morphism $\Delta^k\to \Delta^n$.
I have tried a few things, but not really anything mentionable. Any help is welcome!
edit: as an administrator pointed out, the case $n=1$ is indeed already given as an answer to this question. However, I am still interested in a generalization to arbitrary $n$.