Does category Cat encode all it's properties in itself?

Question

I want specifically know if one can tell that the morphisms between objects in a category are functors or not without being told that the studies category is $Cat$? That means does the objects and morphisms in between them in $Cat$ contain all the information needed to recognize it's morphisms as functors, or such recognition is something external to this category?

Answer 1

The question as it is right know seems to be the following:

Given a category $\mathcal C$, how can I know if $\operatorname{Mor}(\mathcal C) = \operatorname{Mor}(\mathsf{Cat})$?

Then the question does not make much sense. For example, you can probably find a bijection $\phi : \operatorname{Mor}(\mathsf{Cat}) \to \operatorname{Mor}(\mathsf{Set})$ and then craft a new category $\mathcal C$ whose objects are the small sets of and whose morphism class is $\operatorname{Mor}(\mathsf{Cat})$ by setting the sets $\mathcal C(S,T)$ to be $\phi^{-1}(\mathsf{Set}(S,T))$ for each sets $S,T$. So $\mathcal C$ is now a category whose morphism are functors. But what you have done is just renaming each set-function by the name of a functor. It does not tell you anything about $\mathcal C$ or its properties. (And actually, depending on your foundations, this construction might be simply wrong/ill-typed.)

A more sensible question is:

Given a category $\mathcal C$, how can I know if $\mathcal C$ is equivalent to $\mathsf{Cat}$?

Then the question has been tackle by Lawvere under the name Elementary Theory of the Category of Categories (see ETCC).