I'm sure it's clear to most people that there is a connection between the Binomial Theorem and General Leibniz rule, especially when seeing them side by side: $$(a+b)^n=\sum_{k=0}^n {n \choose k} a^k b^{n-k}\longleftrightarrow (f(x)g(x))^{(n)}=\sum_{k=0}^n {n \choose k} f^{(k)}(x) g^{(n-k)}(x),$$
It's been discussed here before: Why General Leibniz rule and Newton's Binomial are so similar?
There, the top answer gives a good example in terms of operators acting on polynomials, but I'd like to get even deeper and more general. I'm sure there's even more examples, which maybe the comments can add, as it would be nice to compile a list.
But in particular, can these be explained from a category-theoretic level? I.e; under what conditions and relationships between some (functors?) $\phi, \psi, \gamma$ obey the most general property that
$$\psi(\phi(X,Y),n)\cong\underset{0\leq k \leq n}{\gamma}\Bigg({n\choose k}, \psi(X, k), \psi(Y, n-k)\Bigg)$$
I.e; in the case of the binomial theorem, we have $$\psi(x, n)=x^n, \phi(a, b)=a+b, \underset{0\leq k \leq n}{\gamma}(x_0, x_1, \cdots) = \sum_{0\leq k \leq n}(x_0 \cdot x_1 \cdots),$$ and in the case of the General Liebniz rule, $$\psi(f, n)=\frac{d^n}{dx^n} f, \phi(a, b)=ab, \underset{0\leq k \leq n}{\gamma}(x_0, x_1, \cdots) = \sum_{0\leq k \leq n}(x_0 \cdot x_1 \cdots).$$
clearly there is some natural transformation between the various places that this "form" arises.
In other words,
Is there some notion of a master theorem in which generalize the Binomial Theorem and General Leibniz rule in more abstract terms?
