homotopy relative to a subset

Let $(X,{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$ be topological spaces, $A\subseteq X$ and $f,g:X\to Y$ continuous maps such that

$$\begin{array}{r}{f}_{|A}=g_{|A}\end{array}$$

Then a homotopy relative to $A$ is a homotopy relative to $f_{|A}$