Composite of homotopies

Definition assuming all homotopies take time $$1$$
Suppose:


 * $$f_1,f_2,f_3:X \to Y$$ are continuous maps
 * $$I$$ is the closed unit interval
 * $$F_{12}:X \times I \to Y$$ is a homotopy from $$f_1$$ to $$f_2$$
 * $$F_{23}:X \times I \to Y$$ is a homotopy from $$f_2$$ to $$f_3$$

Then, the composite of homotopies $$F_{13} = F_{12} * F_{23}$$ is a homotopy from $$f_1$$ to $$f_3$$ given as follows:

$$F_{13}(x,t) = \lbrace\begin{array}{rl}F_{12}(x,2t),& 0 \le t < 1/2\\ F_{23}(x,2t-1), & 1/2 \le t \le 1 \end{array}$$