# Question about homotopy form Hatcher's Algebraic Topology.

In Algebraic topology by Hatcher, there is a statement saying that
A homotopy $$f_t:X\rightarrow X$$ that gives a deformation retraction of $$X$$ onto a subspace $$A$$ has the property that $$f_t(A)=identity$$ for all $$t$$.
I don't understand why this happens. In some steps of deformation retraction the functions $$f_t$$ many not be identical on $$A$$, we only look in the initial and final steps. I have a big confusion about it.
This terminology varies some across authors. Often an author will define a deformation retraction of $$X$$ onto $$A \subset X$$ to be a continuous map $$X \times [0, 1] \to X$$ satisfying (for all $$x \in X$$, $$a \in A$$) $$H(x, 0) = x, \qquad H(x, 1) \in A, \qquad H(a, 1) = a$$ and define a strong deformation retraction to be a deformation retraction that additionally satisfies $$H(a, t) = a$$ for all $$a \in A$$, $$t \in [0, 1]$$ (obviously this condition implies the condition that $$H(a, 1) = a$$ for all $$a \in A$$). But some authors, including Hatcher (see $$\S$$ 0, page 2), define a deformation retraction to satisfy this last condition, too.