In a Category $\textbf{C}$, we say that an $\cal{I}$-diagram in a category is a functor $F: \cal{I} \to \textbf{C}$. In fact, Any object A can be viewed as a diagram by defining a functor $F: \cal{I} \to A$. How would we treat the category then? for example, if I have $\cal{I}$ shaped diagrams in category $\textbf{C}$ would I be able to define the morphisms(natural transformations) between those diagrams(functors)? In other words, should I treat my category as a 2-category now? Would it be the right way? if yes then can I treat every category as a 2-category?
Edit: I am not only talking about cone, which is a constant diagram over $F$. I am also thinking about the natural transformation between diagrams of the same sort that is $F:D \to \textbf{C}$ and $F': D \to \textbf{C}$. In other words category of diagrams (same size and shape). let's say I have triangle-shaped diagrams I am talking about morphisms between them. is there a formal category for that? and how will I treat my original category $\textbf{C}$ now? As a 2-category?