path inside the mapping space as homotopy

Let $X,Y$ be topological spaces, where $X$ is locally compact and $\phi :A\to Y$ a continuous map for $A\subseteq X$. Then a homotopy relative to $\phi$ is precisely given by a path

$$\begin{array}{r}\gamma :[0,1]\to {\mathrm{Map}}_A(X,Y)\end{array}$$

by continuous map and restriction to the image, we may embed w.l.o.g. ${\mathrm{Map}}_A(X,Y)\hookrightarrow \mathrm{Map}(X,Y)$ and thus also

$$\begin{array}{r}\mathrm{Map}([0,1],{\mathrm{Map}}_A(X,Y))\hookrightarrow \mathrm{Map}([0,1],\mathrm{Map}(X,Y))\end{array}$$

(see: covariant compact open functor)

by currying of mapping spaces of locally compact spaces as homeomorphism we then get

$$\begin{array}{r}\mathrm{Map}([0,1],{\mathrm{Map}}_A(X,Y))\cong \mathrm{Map}([0,1]\times X,Y)\end{array}$$

Thus

$$\begin{array}{rlr}\mathrm{Map}([0,1],{\mathrm{Map}}_A(X,Y))\text{arrow}[hookrightarrow]r\iota & \mathrm{Map}([0,1],\mathrm{Map}(X,Y))\text{arrow}[]r\cong & \mathrm{Map}([0,1]\times X,Y)\end{array}$$

We may again restrict the map to the image and get a Bijection

$$\begin{array}{r}\mathrm{Map}([0,1],{\mathrm{Map}}_A(X,Y))\cong {\mathrm{Map}}_{[0,1]\times A}([0,1]\times X,Y)\end{array}$$

Here the restiction is given by the construction of the homeomorphism