Does this specific identity follow from the simplicial identities?
In a simplicial set, if $r<s$, does it follow that the face maps satisfy $$d_s\circ d_r = d_{r+1}\circ d_s?$$ Motivation: I want to show that for all $i<j$, $$d_{n-i}\circ d_{n-j}=d_{n-j+1}\circ d_{n-i},$$ in order to show that for each simplicial set (given by face maps $d_\bullet$, say) the opposite simplicial set (given by face maps $d_{n-\bullet}$) satisfies the simplicial identities too.
Imagine a triangle. Is it true that $d_1\circ d_0 = d_{1}\circ d_1$? Well, $d_1\circ d_1 ([0,1,2]) = d_1 ([0,2]) = [0],$ but $d_0([0,1,2]) = [1,2]$ already does not contain $0.$