The limitation topology and the functor $C(X,Y)$

We consider the category $\mathbf{\Pi}$ where the objects are pairs of topological spaces $(X,Y)$ and the morphisms of a pair $(X,Y)$ to a pair $(E,Z)$ are pairs of continuous maps $(\varphi,\psi)$ where $\varphi :E\mapsto X$, $\psi:Y\mapsto Z$. The space of continuous maps $C_{Lim} (X,Y)$ with the limitation topology defined by H. Torunczyk is assigned to each pair $(X,Y)$. It is proved that this correspondence determines a covariant functor $C_{Lim}$ from the category $\mathbf{\Pi}$ to the category $Top$ of topological spaces with continuous maps. Necessary and sufficient conditions are found to distinguish the subcategory $\mathcal{K}\subset\mathbf{\Pi}$ on which the functor $C_{Lim}$ is continuous.