homotopy relative to a subset

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

\begin{align*} f_{\vert A} = g_{\vert A} \end{align*}

Then a homotopy relative to \(A\) is a homotopy relative to \(f_{\vert A}\)