On the continuity of functors of the type $C(X, Y)$

We consider the category $\mathcal{P}$, the objects of which are pairs of topological spaces $(X, Y)$. Each such pair $(X, Y)$ is assigned the space of continuous maps $C_{\tau}(X, Y)$ with some topology $\tau$. By imposing some restrictions on objects and morphisms of category $\mathcal{P}$, we define a subcategory $\mathcal{K} \subset \mathcal{P}$, for which the above map is a functor from $\mathcal{K}$ to the category Top of topological spaces and continuous maps. The following question is investigated. What are the additional conditions on $\mathcal{K}$, under which the above functor is continuous? Along the way the problem of finding the limit of the inverse spectrum in the category $\mathcal{P}$ is solved. We show, that it reduces to finding the limits of the corresponding direct spectrum and inverse spectrum in the category Top. Point convergence topology, compact-open topology and graph topology are considered as the topology $\tau$.