I suspect that no very satisfying answer to this question is known, so let me just point out that there are a lot of different things one might mean by a "category-theoretic understanding" of a large-cardinal axiom. I am far from an expert, so I hope that others can help flesh this out and correct me, but here are some themes that I happen to have seen in the literature:

1. Set theorists have responded to the paucity of elementary embeddings $V \to V$ by considering elementary embeddings $V \to M$ where $M$ is not $V$. Whereas category theory suggests studying endofunctors $\mathsf{Set} \to \mathsf{Set}$ which satisfy nice properties weaker than being elementary embeddings (actually the first approach can be viewed as a special case of the second, since $M$ is generally assumed to be a non-elementary substructure of $V$, so we can compose $V \to M \subset V$ to get an endofunctor $V \to V$). This approach is much less developed; the only examples I happen to have come across are at the level of measurable cardinals:

• As David Roberts alluded to in the comments, Isbell essentially showed that a full subcategory of $\mathsf{Set}$ is codense iff it is not bounded by any measurable cardinal. A subcategory $A \subseteq B$ is codense (when $B$ is suitably complete) iff its codensity monad (a certain endofunctor $B \to B$, namely the right Kan extension of the inclusion $A \to B$ along itself) is isomorphic to the identity. The codensity monad of a full subcategory $A \subseteq \mathsf{Set}$ is the functor $\mathsf{Set} \to \mathsf{Set}$ sending $X$ to the set of $A$-complete ultrafilters on $X$, hence from a category-theoretic perspective, measurable cardinals can be viewed as connected to the study of certain endofunctors $\mathsf{Set} \to \mathsf{Set}$ which are not elementary embeddings.

• Another relationship between endofunctors of $\mathsf{Set}$ and measurable cardinals is Blass's result that there is a measurable cardinal if and only if there is a left exact endofunctor of $\mathsf{Set}$ not isomorphic to the identity. Both of these results are related to Borger's result that the endofunctor of $\kappa$-complete ultrafilters is terminal among endofunctors of $\mathsf{Set}$ preserving $\kappa$-small coproducts.

2. I'm having a harder time getting a grip on where category-theoretic statements of Vopenka's Principle fit, largely out of ignorance. But it seems like roughly, any ZFC-independent statement about accessible categories is equivalent to Vopenka's Principle, making it hard to see how to generalize the approach. The idea that a large cardinal axiom can be equivalent to a structure axiom about essentially arbitrary classes of first-order models is pretty cool -- I guess this is analogous to having combinatorial statements equivalent to smaller large cardinals. It provides a pretty compelling picture of what Vopenka's Principle means.

3. Type theory has a natural categorical semantics. And it's natural to try to formulate large cardinal axioms in terms of the universes in type theory. There's some work on this by Palmgren, but I think it's dealt mostly with smaller large cardinal axioms.

4. Let me just throw in another possibility that I haven't seen studied, but which could be. One could look for statements high in consistency strength by asking questions about nice functors between categories that don't model ZFC, such as toposes and geometric morphisms. This might go in the same direction as asking about ZF+Reinhardt cardinals, but much further. I have no idea whether this could be a legitimate source of strong statements.

I suspect that no very satisfying answer to this question is known, so let me just point out that there are a lot of different things one might mean by a "category-theoretic understanding" of a large-cardinal axiom. I am far from an expert, so I hope that others can help flesh this out and correct me, but here are some themes that I happen to have seen in the literature:

1. Set theorists have responded to the paucity of elementary embeddings $V \to V$ by considering elementary embeddings $V \to M$ where $M$ is not $V$. Whereas category theory suggests studying endofunctors $\mathsf{Set} \to \mathsf{Set}$ which satisfy nice properties weaker than being elementary embeddings (actually the first approach can be viewed as a special case of the second, since $M$ is generally assumed to be a non-elementary substructure of $V$, so we can compose $V \to M \subset V$ to get an endofunctor $V \to V$). This approach is much less developed; the only examples I happen to have come across are at the level of measurable cardinals:

• As David Roberts alluded to in the comments, Isbell essentially showed that a full subcategory of $\mathsf{Set}$ is codense iff it is not bounded by any measurable cardinal. A subcategory $A \subseteq B$ is codense (when $B$ is suitably complete) iff its codensity monad (a certain endofunctor $B \to B$, namely the right Kan extension of the inclusion $A \to B$ along itself) is isomorphic to the identity. The codensity monad of a full subcategory $A \subseteq \mathsf{Set}$ is the functor $\mathsf{Set} \to \mathsf{Set}$ sending $X$ to the set of $A$-complete ultrafilters on $X$, hence from a category-theoretic perspective, measurable cardinals can be viewed as connected to the study of certain endofunctors $\mathsf{Set} \to \mathsf{Set}$ which are not elementary embeddings.

• Another relationship between endofunctors of $\mathsf{Set}$ and measurable cardinals is Trnková's result (later rediscovered by Blass) that there is a measurable cardinal if and only if there is a left exact endofunctor of $\mathsf{Set}$ not isomorphic to the identity. Both of these results are related to Borger's result that the endofunctor of $\kappa$-complete ultrafilters is terminal among endofunctors of $\mathsf{Set}$ preserving $\kappa$-small coproducts.

2. I'm having a harder time getting a grip on where category-theoretic statements of Vopenka's Principle fit, largely out of ignorance. But it seems like roughly, any ZFC-independent statement about accessible categories is equivalent to Vopenka's Principle, making it hard to see how to generalize the approach. The idea that a large cardinal axiom can be equivalent to a structure axiom about essentially arbitrary classes of first-order models is pretty cool -- I guess this is analogous to having combinatorial statements equivalent to smaller large cardinals. It provides a pretty compelling picture of what Vopenka's Principle means.

3. Type theory has a natural categorical semantics. And it's natural to try to formulate large cardinal axioms in terms of the universes in type theory. There's some work on this by Palmgren, but I think it's dealt mostly with smaller large cardinal axioms.

4. Let me just throw in another possibility that I haven't seen studied, but which could be. One could look for statements high in consistency strength by asking questions about nice functors between categories that don't model ZFC, such as toposes and geometric morphisms. This might go in the same direction as asking about ZF+Reinhardt cardinals, but much further. I have no idea whether this could be a legitimate source of strong statements. EDIT Andreas Blass points out in the comments that there is an example of this at the level of measurable cardinals studied by Adelman and Blass.

I suspect that no very satisfying answer to this question is known, so let me just point out that there are a lot of different things one might mean by a "category-theoretic understanding" of a large-cardinal axiom. I am far from an expert, so I hope that others can help flesh this out and correct me, but here are some themes that I happen to have seen in the literature:

1. Set theorists have responded to the paucity of elementary embeddings $V \to V$ by considering elementary embeddings $V \to M$ where $M$ is not $V$. Whereas category theory suggests studying endofunctors $\mathsf{Set} \to \mathsf{Set}$ which satisfy nice properties weaker than being elementary embeddings (actually the first approach can be viewed as a special case of the second, since $M$ is generally assumed to be a non-elementary substructure of $V$, so we can compose $V \to M \subset V$ to get an endofunctor $V \to V$). This approach is much less developed; the only examples I happen to have come across are at the level of measurable cardinals:

• As David Roberts alluded to in the comments, Isbell essentially showed that a full subcategory of $\mathsf{Set}$ is codense iff it is not bounded by any measurable cardinal. A subcategory $A \subseteq B$ is codense (when $B$ is suitably complete) iff its codensity monad (a certain endofunctor $B \to B$, namely the right Kan extension of the inclusion $A \to B$ along itself) is isomorphic to the identity. The codensity monad of a full subcategory $A \subseteq \mathsf{Set}$ is the functor $\mathsf{Set} \to \mathsf{Set}$ sending $X$ to the set of $A$-complete ultrafilters on $X$, hence from a category-theoretic perspective, measurable cardinals can be viewed as connected to the study of certain endofunctors $\mathsf{Set} \to \mathsf{Set}$ which are not elementary embeddings.

• Another relationship between endofunctors of $\mathsf{Set}$ and measurable cardinals is Blass's result that there is a measurable cardinal if and only if there is a left exact endofunctor of $\mathsf{Set}$ not isomorphic to the identity. Both of these results are related to Borger's result that the endofunctor of $\kappa$-complete ultrafilters is terminal among endofunctors of $\mathsf{Set}$ preserving $\kappa$-small coproducts.

2. I'm having a harder time getting a grip on where category-theoretic statements of Vopenka's Principle fit, largely out of ignorance. But it seems like roughly, any ZFC-independent statement about accessible categories is equivalent to Vopenka's Principle, making it hard to see how to generalize the approach. The idea that a large cardinal axiom can be equivalent to a structure axiom about essentially arbitrary classes of first-order models is pretty cool -- I guess this is analogous to having combinatorial statements equivalent to smaller large cardinals. It provides a pretty compelling picture of what Vopenka's Principle means.

3. Type theory has a natural categorical semantics. And it's natural to try to formulate large cardinal axioms in terms of the universes in type theory. There's some work on this by Palmgren, but I think it's dealt mostly with smaller large cardinal axioms.

I suspect that no very satisfying answer to this question is known, so let me just point out that there are a lot of different things one might mean by a "category-theoretic understanding" of a large-cardinal axiom. I am far from an expert, so I hope that others can help flesh this out and correct me, but here are some themes that I happen to have seen in the literature:

1. Set theorists have responded to the paucity of elementary embeddings $V \to V$ by considering elementary embeddings $V \to M$ where $M$ is not $V$. Whereas category theory suggests studying endofunctors $\mathsf{Set} \to \mathsf{Set}$ which satisfy nice properties weaker than being elementary embeddings (actually the first approach can be viewed as a special case of the second, since $M$ is generally assumed to be a non-elementary substructure of $V$, so we can compose $V \to M \subset V$ to get an endofunctor $V \to V$). This approach is much less developed; the only examples I happen to have come across are at the level of measurable cardinals:

• As David Roberts alluded to in the comments, Isbell essentially showed that a full subcategory of $\mathsf{Set}$ is codense iff it is not bounded by any measurable cardinal. A subcategory $A \subseteq B$ is codense (when $B$ is suitably complete) iff its codensity monad (a certain endofunctor $B \to B$, namely the right Kan extension of the inclusion $A \to B$ along itself) is isomorphic to the identity. The codensity monad of a full subcategory $A \subseteq \mathsf{Set}$ is the functor $\mathsf{Set} \to \mathsf{Set}$ sending $X$ to the set of $A$-complete ultrafilters on $X$, hence from a category-theoretic perspective, measurable cardinals can be viewed as connected to the study of certain endofunctors $\mathsf{Set} \to \mathsf{Set}$ which are not elementary embeddings.

• Another relationship between endofunctors of $\mathsf{Set}$ and measurable cardinals is Blass's result that there is a measurable cardinal if and only if there is a left exact endofunctor of $\mathsf{Set}$ not isomorphic to the identity. Both of these results are related to Borger's result that the endofunctor of $\kappa$-complete ultrafilters is terminal among endofunctors of $\mathsf{Set}$ preserving $\kappa$-small coproducts.

2. I'm having a harder time getting a grip on where category-theoretic statements of Vopenka's Principle fit, largely out of ignorance. But it seems like roughly, any ZFC-independent statement about accessible categories is equivalent to Vopenka's Principle, making it hard to see how to generalize the approach. The idea that a large cardinal axiom can be equivalent to a structure axiom about essentially arbitrary classes of first-order models is pretty cool -- I guess this is analogous to having combinatorial statements equivalent to smaller large cardinals. It provides a pretty compelling picture of what Vopenka's Principle means.

3. Type theory has a natural categorical semantics. And it's natural to try to formulate large cardinal axioms in terms of the universes in type theory. There's some work on this by Palmgren, but I think it's dealt mostly with smaller large cardinal axioms.

4. Let me just throw in another possibility that I haven't seen studied, but which could be. One could look for statements high in consistency strength by asking questions about nice functors between categories that don't model ZFC, such as toposes and geometric morphisms. This might go in the same direction as asking about ZF+Reinhardt cardinals, but much further. I have no idea whether this could be a legitimate source of strong statements.