This note is based on Hu's Homotopy Theory (New York, 1959), pp. 115 f.
The theorem that the homotopy sequence is exact splits into six statements. The fifth in Hu's exposition can be stated as follows: Let f:In, In-1, Jn-1→X, A, x0,
Then there exists a map
such that f~j g: In, In-1, Jn-1→X, A, x0.