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.
Yes, $i$ and $j$ are not allowed to be $k$ in any of the terms here (for confirmation, compare the statement of Corollary 3.2). Note that the top and bottom rows the diagram are also unnecessary (they won't change the colimit since they are just included into the middle row) and appear to be included only to clarify what the maps in the middle row are (namely, they are the maps that will make the diagram commute with the top and bottom rows).