To determine the nature of the mathematical structure \((\mathcal{F}, \circ)\) where \(\mathcal{F}\) is the set of functions from set \(A = \{0, 1, 2, 3, \ldots\}\) to itself and the operation \(\circ\) is defined as:

\((f_1 \circ f_2)(n) = f_1(n) + f_2(n)\)

we need to check whether it satisfies the properties of various algebraic structures such as groups and monoids. Let's go through these properties one by one.

1. Closure: For any two functions \(f_1, f_2 \in \mathcal{F}\), the sum \(f_1(n) + f_2(n)\) is a non-negative integer for any \(n \in A\). Therefore, the result is still a function from \(A\) to itself. Thus, closure under the operation \(\circ\) is satisfied.
2. Associativity: The operation \((f_1 \circ f_2) \circ f_3 = f_1 \circ (f_2 \circ f_3)\) holds, as addition of integers is associative. Therefore, associativity is satisfied.
3. Identity Element: The identity function \(e(n) = 0\) for all \(n \in A\) acts as an identity element because \((f \circ e)(n) = f(n) + 0 = f(n)\) and \((e \circ f)(n) = 0 + f(n) = f(n)\) for any function \(f \in \mathcal{F}\).
4. Invertibility: To be a group, each function \(f\) should have an inverse such that \((f \circ g)(n) = e(n) = 0\) for some function \(g\). However, for a function \(f(n) \neq 0\), there is no function \(g\) such that \(f(n) + g(n) = 0\) for all \(n\) because all elements must be non-negative integers. Therefore, invertibility is not possible for any \(f \neq e\).
5. Commutativity: Since addition is commutative, \((f_1 \circ f_2)(n) = f_1(n) + f_2(n) = f_2(n) + f_1(n) = (f_2 \circ f_1)(n)\), the operation is commutative.

With these properties evaluated, we find that \((\mathcal{F}, \circ)\) satisfies closure, associativity, has an identity element, and is commutative, but does not satisfy invertibility. Hence, it is an Abelian Monoid.

Therefore, the correct answer is: