Quantum Functor $Mor$

Let $Top_c$ be the category of compact spaces and continuous maps and $Top_fsubset Top_c$ be the full subcategory of finite spaces. Consider the covariant functor $Mor:Top_f^{op}times Top_cto Top_c$ that associates any pair $(X,Y)$ with the space of all morphisms from $X$ to $Y$. In this paper, we describe a non commutative version of $Mor$. More pricelessly, we define a functor $mathfrak{M}mathfrak{o}mathfrak{r}$, that takes any pair $(B,C)$ of a finitely generated unital C*-algebra $B$ and a finite dimensional C*-algebra $C$ to the quantum family of all morphism from $B$ to $C$.