If $X,Y$ are pointed spaces in $\text{hTop}_*$, then what is the relationship between $[\Sigma X, Y]$ and $[X \times I, Y]$, where $\Sigma$ means the reduced suspension and the brackets mean the set of all continuous functions from $()$ to $()$?
Indication suggests that there's a bijection between them but I can't see how this could be since quotient spaces aren't in one-to-one correspondence to the spaces they're derived from.
As $X\times I$ is homotopy equivalent to $X$ relative basepoints, $[X\times I, Y] = [X, Y]$. There is a natural map $i : X \to \Sigma X$, given by $x \mapsto [(x, \frac{1}{2})]$, which induces a map $[\Sigma X, Y] \to [X, Y]$ given by precomposition with $i$. However, there is no stronger relation between the two. For example, if $X = S^n$, then $[X\times I, Y] = \pi_n(Y)$ and $[\Sigma X, Y] = \pi_{n+1}(Y)$. In particular, the map $[\Sigma X, Y] \to [X, Y]$ need not be injective nor surjective.