 Solution

For two neighboring comoving points separated along the zeroth-order direction $\hat{\mathbf{n}}$ by coordinate distance $\delta \chi$, the synchronous gauge spatial metric gives the cosmological proper distanceDifferentiate with respect to cosmic time at fixed coordinate separation. To first order,A photon crosses this neighboring interval in $\delta t=\delta r$ when $c=1$, so its local relative recession velocity isAt each infinitesimal step, the local Doppler effect gives $d\log \nu =-\Delta v$. ThereforeThe first term integrates to the homogeneous cosmological redshift, $\bar{\nu }\propto a^{-1}$. Subtracting it isolates the metric contribution to the fractional frequency shift. A shifted blackbody radiation spectrum has its temperature shifted by the same fraction as its photon frequencies, givingThe dot denotes a partial time derivative at fixed comoving position, evaluated along the zeroth-order null geodesic. It is not the total derivative of $h_{ij}$ along the ray, so the integral cannot in general be replaced by a difference of $h_{ij}$ at the endpoints. Deflection of the ray or its direction multiplies an already first-order metric perturbation and is second order here. This is the metric part of the Sachs-Wolfe effect in the negative-spatial-perturbation convention; an intrinsic emission temperature fluctuation or an emitter velocity would add separate boundary terms. The time derivative is present in the original PDF, although missing from the converted TeX's displayed integral.