categorical characterization of large cardinals Question 1. Is there a categorical representation of Kunen's inconsistency result? 
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and supercompact cardinals)?
Question 3.  Is there a categorical characterization of $0^{\sharp}$?
Remark 1. A categorical characterization of other large cardinals is also welcome. 
Remark 2.  Andreas Blass in the paper "Exact functors and measurable cardinals" has proved that the existence of a measurable cardinal is equivalent to the existence of a non-trivial exact functor from the category of sets to the category of sets.
Would you please give references for such matters.
Remark 3. The following papers may have some information about the relation between category theory and large cardinals:
1) Adequate subcategories-Isbell
2) Small adequate subcategories-Isbell,
3) Structure of categories-Isbell,
4) Exact functors and measurable cardinals-Blass,
5) Exact functors, local connectedness and measurable cardinals-Adelman & Blass.
 A: In the paper Proof theory and set theory Takeuti has given such a characterization for measurable cardinals, strongly compact cardinals, supercompact cardinals and even large cardinals.
Let me first  give the characterization for what Takeuti calls an $\omega$-huge cardinal.  Call $\kappa$ is $\omega$-huge if there exists a non-trivial elementary embedding $j: V_\gamma \to V_\gamma$ with critical point $\kappa$ such that $\gamma=\sup_{n<\omega}\kappa_n$ with $\kappa_0=\kappa$ and
$\kappa_{n+1}=j(\kappa_n)$.
Let $\mathcal{C}$ be the category whose objects are the sets $V_\alpha,$ $\alpha$ an ordinal and whose arrows are elementary embeddings. Then the following are equivalent:

*

*There is an $\omega$-huge cardinal,


*there exists a functor $F: \mathcal{C} \to \mathcal{C}$ and a nontrivial natural transformation
$\eta: F \to F.$
Now let give a characterization of measurable cardinals. $\kappa$ is measurable iff there exists a functor $F:\bf Set \to Set$ such that:

*

*$F$ commutes with direct limits whose cardinality is $<\kappa,$


*$F$ commutes with pullbacks and pushouts, and


*$F$ does not commute with the following simplest direct limit with the cardinality
$\kappa$:
$F(\kappa) \neq \operatorname{dirlim} (F(\alpha), (F(i_{\alpha, \beta}))_{\alpha \leq \beta <\kappa}$,
where $i_{\alpha, \beta}$ is the identity map from $\alpha$ to $\beta.$
