Some topological properties of the functor $C(X,Y)$

Let $\mathcal{K}$ is a category which objects are some pairs of topological spaces $(X,Y).$ To every pair $(X,Y)$ we associate a space $C_\tau(X,Y)$ of continuous maps endowed with some topology $\tau,$ and to every morphism $\mathcal{K}$ corresponding connecting maps between $C\tau(X,Y)$ spaces. In this paper we study the possibility of the defined map from $\mathcal{K}$ to category $Top$ of topological spaces and continuous maps to be a functor and its continuity.