homotopy relative to a map

Let \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) be topological spaces, \(\varphi: A \rightarrow Y\) a continuous map for \(A \subseteq X\). Suppose \(f,g \in \mathcal{C}_{\varphi}(X,Y)\) are elements in the relative mapping set.

Then a homotopy relative to \(\varphi\) from \(f\) to \(g\) is a continuous map

\begin{align*} H: X \times [0,1] \rightarrow Y \end{align*}

such that

1. \(H(x,0) = f(x)\) for \(x \in X\)
2. \(H(x,1) = g(x)\) for \(x \in X\)
3. $H(a,t) = ϕ(a)& for \(a \in A\)