Why is this true? $L\left[\int_{a}^{b}G(x,s)g(s) ds\right] = \int_{a}^{b}L[G(x,s)]g(s) ds?$

Question

I'm learning how Green's functions are used to solve second order inhomogeneous ODEs.

If $$L[y] = a_2(x)y'' + a_1(x)y' + a_0(x)y,$$ how come $$L\left[\int_{a}^{b}G(x,s)g(s) ds\right] = \int_{a}^{b}L[G(x,s)]g(s) ds?$$

I understand why this is the case if $G(x,s) = f(x)q(s)$, but struggle to convince myself when this isn't the case. Is there any good way of thinking about this?