Let $f:\Delta\to \Delta$ be the functor given by $[n]\mapsto [n]\star[n]=[2n+1]$. We can extend $f$ cocontinuously to a functor $$f_!: SSet\to SSet$$ (that is, the left adjoint of the functor $f^*$. Notice that $f_!S \neq S\star S$ in general.)
My question: Does $f_!$ send (inner) anodyne maps to (inner) anodyne maps? When I draw a picture, then it seems quite sure that the answer should be true.