If $X, Y$ are topological spaces, let $\newcommand{\Cont}{\text{Cont}}\Cont(X,Y)$ denote the collection of continous maps $f:X\to Y$, and we endow $\Cont(X,Y)$ with the product topology inherited from $Y^X$.

Question. Given a topological space $X$, can we always find a topological space $Z$ such that $\Cont(X,Z) \cong X$?

If $X, Y$ are topological spaces, let $\newcommand{\Cont}{\text{Cont}}\Cont(X,Y)$ denote the collection of continous maps $f:X\to Y$, and we endow $\Cont(X,Y)$ with the product topology inherited from $Y^X$.

Question. Is there a topological space $(X,\tau)$ with $\tau\neq\{\emptyset, X\}$ such that there is a topological space $Z$ with $\Cont(X,Z) \cong X$?