Can anyone please help me with this question?
For any family of triple of spaces $(X_i,B_i,A_i | i \in I)$, show that there is a bijection $$\left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), \prod (X_i,B_i,A_i)\right] \rightarrow \prod\left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), (X_i,B_i,A_i)\right]$$
The key point of this question is to understand what are the elements of each set involved. The elements of $\left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), \prod (X_i,B_i,A_i)\right]$ are homotopy classes $[G]$ of maps $$ G:\mathbb{D}^n\rightarrow \prod_iX_i::x\mapsto (G_i(x))_i, $$ where $G_i:\mathbb{D}^n\rightarrow X_i$ is the $i$-th component of $G$. On the other hand, elements of $\left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), (X_i,B_i,A_i)\right]$ are sequences $([f_i])_i$ of homotopy classes of maps $$ f_i:\mathbb{D}^n\rightarrow X_i. $$ With this premise you can define a function $$ h:\left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), \prod (X_i,B_i,A_i)\right] \rightarrow \prod\left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), (X_i,B_i,A_i)\right]$$ by $$ h([(G_i)_i]):=([G_i])_i $$ The function is well-defined. Indeed, $[(G_i)_i]=[(F_i)_i]$ implies $[G_i]=[F_i]$ for all $i$.
Furthermore, it is injective. Indeed, $\ker(h)=h^{-1}(\{([0])_i\})=[(0)_i]$.
Finally, it is surjective. Indeed, given $([f_i])_i\in\prod \left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), (X_i,B_i,A_i)\right]$, we have $$ ([f_i])_i=h([F]), $$ where $$F:=(f_i)_i:\mathbb{D}^n\rightarrow \left[(\mathbb{D^n, \mathbb{S^{n-1}}}, *), \prod (X_i,B_i,A_i)\right]. $$