I have to admit not having ever seen a proof of what Max was asking about. Here is the way I see it in dimension 1. I assume that it is similar, but more complicated in higher dimensions. Consider the beginnings of a simplicial set and assume that homotopy, which I will denote ~ is an ER: d^0 ----> X_1 <---- X_0 ----> d^1 with the degeneracy s in the middle. The presence of the degeneracy forces the homotopy relation to be reflexive, by the way, although that is not what is involved here, but it is what is behind Max' question. The crucial thing is that any 1-cell is a homotopy between its two vertices. If x in X_1, then x is a homotopy between d^0(x) and d^1(x). Now let x^0, x^1 in X_1 satisfy d^0(x^0) = d^0(x^1), which is the only equation satisfied by that kind of 1-horn. Then we have d^0(x^0) ~ d^1(x^0), d^0(x^0) ~ d^1(x^0) and d^0(x^0) = d^0(x^1). If homotopy is to be an ER, we must have d^1(x^0) ~ d^1(x^1), which means that we need an element we will call x^2 in X_1 such that d^0(x^2) = d^1(x^0) and d^1(x^2) = d^1(x^1), which is the fill-in required. There are two other kinds of horns, depending on which simplex is omitted, but the one with x^1 omitted I have checked and it is similar and the third one is just the reverse of this one. This must be in some of the early papers of Kan (late 50s) or maybe John Moore (early 60s). Don't ask the categories net; ask algebraic topologists. Michael ======================================