Let $F:I \to C$ be a functor, where $I$ is an index category and $C$ is a category. Show there is a natural bijection between $\operatorname{Mor}_C(T, \lim_{i \in I}F(i))$ and $\lim_{i \in I} \operatorname{Mor}_{C}(T,F(i))$.
What is the meaning of $\lim_{i \in I} \operatorname{Mor}_{C}(T,F(i))$ ? Does this notation imply limits on the category of morphisms of $C$ or category of sets?
It is a limit computed in $Set$, precisely of the functor $Hom_{\mathscr{C}}(T,-) \circ F(-):I \to Set$.