In this post (Is there a categorification of (infinite) ordinal arithmetic?) the answer by Martin Brandenburg gives a nice category theoretic characterization of ordinal sums and (binary) ordinal products, and then states that ordinal exponentiation can be characterized similarly.
However, when trying to dualize the definitions he used for the "directed coproducts" I seem to run into trouble: for the simplest case, suppose $S$ is the ordinal number $2=\{0,1\}$ and we have well orderings $X_0$ and $X_1$. Then, we should expect to obtain the anti-lexicographic order on the set product $X_0\times X_1$ for our "directed product" $X_0\otimes X_1$ to match with the usual definition on ordinals. That is, we can consider the projection maps $\pi_i:X_0\otimes X_1\to X_i$ and for any collection of maps $f_i:Y\to X_i$ satisfying [some property dual to that in the linked answer], there should be a unique map $f:Y\to X$ with $\pi_i\circ f=f_i$.
The first problem I run into is that it isn't obvious what if any relation we should require $f_0$ and $f_1$ to have relative to one another: they do not share a codomain and so I don't see a nice way to compare their values. More importantly, the map $\pi_0:X_0\otimes X_1\to X_0$ is not even order-preserving, and thus is not a morphism in the category being considered, except in trivial cases. Given $a<b$ in $X_0$ and $c<d$ in $X_1$ we have $(b,c)<(a,d)$ in $X_0\otimes X_1$ but $\pi_0(b,c)=b>a=\pi_0(a,d)$.
Is there a way around the above two issues? If not, is there some other way to categorically characterize ordinal exponentiation and (non-binary) products? I'm really more interested in using this to come up with an analogous categorical notion of "product indexed by an order" in the larger category of Partial Orders, but I suspect an answer in the special case of well-orderings should be generalizable.