I am having trouble understanding the maps in this coequalizer defining the k-horn (from page 9 of Goerss and Jardine's Simplicial Homotopy Theory). It is defining them using the ith, jth inclusion into the coproduct over all indices not equal to k. However, i, j do not have this restriction, so I don't think the maps described make sense when i or j equals k.
Am I correct that this is an issue? Is this fixed by just requiring i, j to not be k? That makes the most sense to me geometrically.