Convolution with Dirac comb

I am trying to understand convolutions with distributions involving sums of delta functions. Consider the Dirac comb $\text{Sha} = \sum_{n \in \mathbb{Z}} \delta(x-n)$ and a function $f(x)$.

The convolution $(\text{Sha} * f)(x)$ is formally:$$\langle \sum_{n \in \mathbb{Z}} \delta(y-n), f(x-y) \rangle = \sum_{n \in \mathbb{Z}} f(x-n)$$

My question is about the interchange of sum and distributional pairing in this equation. Is this interchange something that needs to be justified, or does it simply follow directly from the definition of how delta functions act on test functions?

More precisely:

1. Is the equation $(\text{Sha} * f)(x) = \sum_{n \in \mathbb{Z}} f(x-n)$ simply the definition of how convolution with a sum of deltas works?
2. Or do we need to justify interchanging the sum and the distributional pairing $\langle \cdot, \cdot \rangle$?\end{enumerate}

I would appreciate understanding this for both:

a. Nice functions $f$ (e.g., Schwartz class)
b. More general cases where $\sum_{n \in \mathbb{Z}} f(x-n)$ is only locally integrable

The broader context is that I'm studying convolutions with weighted sums of delta functions, and I want to understand when we can write$$\left(\sum_{n} c_n \delta(x-a_n)\right) * f = \sum_{n} c_n f(x-a_n)$$without additional justification.