In category theory there are definitions for $A\oplus B$, $A\times B$ and $A^B$ via universal properties. I wonder if it is possible to isolate a particular universal property to represent the tetration of $A,B$ which we denote it by $A\uparrow B$. Intuitively $A\uparrow B$ is $\underbrace {A^{A^{A^{.^{.^{.}}}}}}_{B-times}$.
Question: What is the category theoretic definition of $A\uparrow B$ object?
Remark: Regarding the comments on finding some examples of tetration of two mathematical objects, I think this is exactly the difficulty of the problem. It seems there is no intuition about tetration and other hyperoperators out of number theory. But I think there is an "implicit" way to describe such an object in category theory via the notions of "exponentiation" and "limit" objects. In fact I hope one may give me a purely abstract way of defining tetration of two objects via categorical constructions that could be used as a base of definition for tetration of two objects in different contexts.
Let $a,b$ be objects of a closed symmetric monoidal category. Then $a^b$ may be written for the internal hom $\underline{\hom}(b,a)$. In fact, then we have the usual laws such as $a^{b+c}=a^b \times a^c$ and $(a^b)^c = a^{b \times c}$.
Now let us iterate this. $a^a = \underline{\hom}(a,a)$, $a^{a^a} = \underline{\hom}(\underline{\hom}(a,a),a)$, etc. We can define $^{n} a=a^{a^{a^{a^\dotsc}}}$ for every $n < \omega$. Assume that every object is dualizable (for example, consider the category of finite-dimensional vector spaces over some field), so that $a^b = b^* \otimes a$. Then one shows by induction that $$^n a = \left\{\begin{array}{c}(a^*)^{\otimes \frac{n}{2}} \otimes a^{\otimes \frac{n}{2}} & n \text{ even} \\ (a^*)^{\otimes \frac{n-1}{2}} \otimes a^{\otimes \frac{n+1}{2}} & n \text{ odd}\end{array}\right.$$ This case distinction indicates that it is impossible to give a natural definition of $^b a$ for objects $a,b$.