= Solution
Along the zeroth-order photon path, set $Y=\Theta+\psi$ and $\mathcal S=(3/4)\int d\hat{\mathbf m}\,\Theta(\hat{\mathbf m})[1+(\mathbf e\cdot\hat{\mathbf m})^2]/(4\pi)$. The <scalar photon energy redshift> and the <Photon Boltzmann equation with Thomson scattering> give
$$
\frac{dY}{d\eta}=\phi'+\psi'+\tau'\left[Y-\psi-\mathcal S-\mathbf e\cdot\mathbf v_b\right].
$$
The integrating factor is $e^{-\tau}$, because $\tau'<0$ in this convention. Hence
$$
\frac d{d\eta}(e^{-\tau}Y)=e^{-\tau}(\phi'+\psi')
+g(\eta)(\psi+\mathcal S+\mathbf e\cdot\mathbf v_b),\quad g=-\tau'e^{-\tau}.
$$
<Tight coupling> before decoupling makes the incident distribution nearly isotropic in the baryon frame. Neglecting its quadrupole, the angular kernel integrates to $\mathcal S=\Theta_0$, since $(3/4)\langle1+(\mathbf e\cdot\hat{\mathbf m})^2\rangle=1$. The primordial term at the optically thick lower endpoint is erased. With instantaneous last scattering, no later scattering and $g=\delta(\eta-\eta_*)$, \b[the line-of-sight solution is]
$$
\boxed{\Theta(\eta_0,\mathbf x_0,\mathbf e)+\psi(\eta_0,\mathbf x_0)
=\left(\Theta_0+\psi+\mathbf e\cdot\mathbf v_b\right)(\eta_*,\mathbf x_0-\chi_*\mathbf e)
+\int_{\eta_*}^{\eta_0}d\eta\,(\phi'+\psi')(\eta,\mathbf x_0-(\eta_0-\eta)\mathbf e).}
$$
The approximation is linear, uses an unperturbed ray, neglects the scattering quadrupole and polarization feedback, and idealizes the <cosmological visibility function> as a delta function. It also excludes <reionization> or any other post-recombination opacity.
The <photon monopole> $\Theta_0$ is the intrinsic emission-temperature perturbation. The emission potential $\psi_*$ gives gravitational redshift on leaving the potential well; together they form the <Sachs-Wolfe combination>. The baryon <peculiar velocity> supplies the <Doppler CMB anisotropy>. The integral is the <Integrated Sachs-Wolfe effect>, from changes in gravitational potentials during travel. The observer potential is a direction-independent endpoint shift absorbed into the observed monopole. An observer <peculiar velocity>, if included, contributes an additional kinematic dipole; the stated observer is coordinate-comoving.
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