Universal property of functor category Between two categories $\mathcal{C}$ and $\mathcal{D}$ there is a functor category $Fun(\mathcal{C},\mathcal{D})$ with functors as objects and natural transformations as morphisms.   
Is there a universal property that can be used to define a functor category (up to equivalence)?  
 A: Any category can be a functor category, because let $\mathcal C:=\{1\}$ the trivial category with one object, then $Fun(\mathcal C,\mathcal D)\simeq \mathcal D$.
A: Here is a functor category that has a 2-universal property:
Theorem. Let $\mathbb{C}$ be a small category and let $h_\bullet : \mathbb{C} \to [\mathbb{C}^\textrm{op}, \textbf{Set}]$ be the Yoneda embedding. Then, for all locally small and cocomplete categories $\mathcal{E}$, the functor $F \mapsto F h_\bullet$ from the category of cocontinuous functors $[\mathbb{C}^\textrm{op}, \textbf{Set}] \to \mathcal{E}$ to the category of all functors $\mathbb{C} \to \mathcal{E}$ is fully faithful and essentially surjective on objects, and this functor is pseudonatural in $\mathcal{E}$. In other words, $[\mathbb{C}^\textrm{op}, \textbf{Set}]$ is the free cocompletion of $\mathbb{C}$.
But more generally, if $\mathbb{C}$ and $\mathbb{D}$ are both small categories, then $[\mathbb{C}, \mathbb{D}]$ has a 1-universal property:
Theorem. There is a bijection between functors $\mathbb{E} \times \mathbb{C} \to \mathbb{D}$ and functors $\mathbb{E} \to [\mathbb{C}, \mathbb{D}]$ and this bijection is natural in $\mathbb{C}, \mathbb{D}, \mathbb{E}$. In other words, the category of small categories is cartesian closed and the functor category is an exponential object.
A: Amplifying Zhen's answer. One may show, that $[\mathbb{C}, \mathbb{D}]$ satisfies the 2-universal property, i.e. there exists a 2-natural isomorphism of categories:
$$\hom(\mathbb{E} \times \mathbb{C}, \mathbb{D}) \approx \hom(\mathbb{E}, [\mathbb{C}, \mathbb{D}])$$
making $\mathbf{Cat}$ a 2-cartesian closed category. Moreover this structure is the unique 2-closed symmetric monoidal structure on $\mathbf{Cat}$.
