Suppose $X$ and $Y$ are topological spaces and $Y$ is a based space with basepoint $b$. The (unpointed) hom space $[X,Y]$ has a natural choice of basepoint, i.e. $c_b$, the constant map sending everything to $b$ - so we can consider the loop space $\Omega[X,Y]$. Is it true that there's a homeomorphism $$\Omega[X,Y] \cong [X, \Omega Y]$$
The comments in the answer of the post here seem to suggest so, but I don't see why.