Problem given. Let $(X_i)_{i\in I}$ and $(Y_i)_{i\in I}$ be topological spaces. Assume $X_i$ is homotopy equivalent to $Y_i$. Show that or provide a counterexample to: $\coprod_{i\in I}X_i$ is homotopy equivalent to $\coprod_{i\in I}Y_i$.
Does not this follow because if $f_i:X_i\rightarrow Y_i$, $g_i:Y_i\rightarrow X_i$ are such that $f_i\circ g_i\simeq id$ and $g_i\circ f_i\simeq id$ then we get $f:\coprod_i X_i\rightarrow \coprod_i Y_i$ , $g:\coprod_iY_i\rightarrow \coprod_i X_i$ that are homotopy equivalent? (Our topologylecturer calls this property the 'universal property' of coproduct)