# Need help understanding mathematical step in QFT book

I am reading Quantum Field Theory, by Mark Srednicki. On page 50, he derives the scattering amplitude for two spinless particles from $\pm \infty$. I am stuck on what I believe is a simple thing. His steps are the following:

\begin{align} a_1^{\dagger}(+\infty)-a_1^{\dagger}(-\infty) &= \int_{-\infty}^{\infty}dt \partial_0 a_1^{\dagger}(t) \\ &= -i\int d^3k f_1(\mathbf{k})\int d^4 x \partial_0 \left( e^{ikx}\overset\leftrightarrow\partial_0 \psi(x) \right) \\ &= -i\int d^3k f_1(\mathbf{k})\int d^4 x e^{ikx}\left( \partial_0^2+\omega^2 \right)\psi(x) \\ &= -i\int d^3k f_1(\mathbf{k})\int d^4 x e^{ikx}\left( \partial_0^2+\mathbf{k}^2 + m^2 \right)\psi(x) \\ &= -i\int d^3k f_1(\mathbf{k})\int d^4 x e^{ikx}\left( \partial_0^2-\overset\leftarrow\nabla^2 + m^2 \right)\psi(x) \\ &= -i\int d^3k f_1(\mathbf{k})\int d^4 x e^{ikx}\left( \partial_0^2-\overset\rightarrow\nabla^2 + m^2 \right)\psi(x) \\ &= -i\int d^3k f_1(\mathbf{k})\int d^4 x e^{ikx}\left( -\partial^2 + m^2 \right)\psi(x) \\ \end{align}

And here is the explanation for these steps. What I don't understand is highlighted in bold.

The first equality is just the fundamental theorem of calculus. To get the second, we substituted the definition of $a^{\dagger}_1(t)$, and combined the $d^3x$ from this definition with the $dt$ to get $d^4x$. The third comes from straightforward evaluation of the time derivatives. The fourth uses $ω^2 = \mathbf{k}^2+m^2$. The fifth writes $k^2$ as $−∇^2$ acting on $e^{ik·x}$. The sixth uses integration by parts to move the $∇^2$ onto the field $\psi(x)$; here the wave packet is needed to avoid a surface term. The seventh simply identifies $∂^2_0 − ∇^2$ as $−∂^2$.

What is the difference between the del with the right arrow and the left arrow? And how does integration by parts change one to the other?

Before integration by parts: $$-e^{ikx} (\overleftarrow{\nabla}^2) \psi(x) = -(\nabla^2 e^{ikx}) \psi(x).$$
Integrate by parts once: $$+(\nabla e^{ikx}) \cdot (\nabla \psi(x)) + \text{boundary terms.}$$
And a second time: $$-e^{ikx}\, \nabla^2 \psi(x) + \text{boundary terms.}$$