Are there standard names for the ends of a homotopy?

Question

In my mind, homotopies have a beginning and an end, a directionality, it is always from something to something.

Are there standard names for these in the literature? Can ppl be expected to understand if I write "the end of the homotopy $h$ is..."?

Answer 1

Ronald Brown, Topology and Groupoids (2006), p.225:

Let $X, Y$ be topological spaces. A map $F \colon X \times [0, q] \to Y$ will be called a homotopy of length $q;$ for such $F,$ the initial map and the final map of $F$ are respectively the functions $$ \begin{aligned} f \colon X & \to Y \\ x & \mapsto F(x, 0) \end{aligned} \qquad \begin{aligned} g \colon X & \to Y \\ x & \mapsto F(x, q). \end{aligned} $$ We say $F$ is a homotopy from $f$ to $g$, and we write $$ F \colon f \simeq g. $$