Puppe sequence ($A\to X\to C_i\to \Sigma A\to \Sigma X\to \cdots$)

Successive mapping cones of a based map give this sequence up to homotopy. Applying based maps into a space gives an exact sequence of pointed sets in the opposite direction. The exactness statement at $[X,Y]$ says that a map whose restriction to $A$ is null-homotopic extends over $C_i$ using the chosen null-homotopy.