category cat as cartesian closed category

Let \(\mathrm{Cat}\) be the Category Cat.
Then \(\mathrm{Cat}\) is cartesian closed category via

\begin{align*} X^Y = \mathrm{Fun}(Y,X) \end{align*}

see:

• functor category
  
• large cardinal axiom
  

• category cat as complete category
  
• currying in category cat