Where does the matching condition for $U(1)$ subgroups come from in unified models? The matching conditions for a breaking $G \rightarrow \prod_i G_i$ are
$$\omega_G-\frac{C_2(G)(\mu)}{12 \pi}=\omega_{G_i}-\frac{C_2(G_i)(\mu)}{12 \pi} ,$$
where $C_2(g)$ denotes the quadratic Casimir invariant for the adjoint representation of the group $g$. (See, for example, Eq. 7 in  Implications of the CERN LEP results for SO(10) grand unification )
However, for example, for the braking chain $SU(4) \times SU(2)_L \times  SU(2)_R \rightarrow SU(3_C) \times   SU(2)_L \times U(1)_Y$ we have the matching condition
$$ \omega_{U(1)_Y}  =  \frac{3}{5}\left( \omega_{SU(2)_R} - \frac{2}{12\pi}  \right) + \frac{2}{5}\left( \omega_{SU(4)_C} - \frac{4}{12\pi}  \right)$$
(See, for example, Eq. 8 in  Implications of the CERN LEP results for SO(10) grand unification )
Where does this matching condition for $U(1)$ subgroups come from?
 A: It results from the combination of two facts: $i$) the embedding of $U(1)_Y$ into $SU(2)_R \times U(1)_{B-L}$, and $ii$) the normalization of the $U(1)$ charges.
Let me take the simpler chain: $SU(2)_L \times SU(2)_R \times U(1)_{B-L} \to SU(2)_L \times U(1)_Y$.
$i$)
At the scale of the left-right symmetry, the hypercharge gets merged into $SU(2)_R \times U(1)_{B-L}$, and we have the relation: 
\begin{equation}
\frac{Y}{2} = T_{3R} + \frac{B-L}{2}\,.
\end{equation} 
(the factors of 1/2 here are conventional). Here $T_{3R}$ is the generator of $SU(2)_R$.
$ii$)
In order for the non-abelian generators to be normalized like their abelian counterparts, we define the new charges:
$Y' = \sqrt{\frac{3}{5}} \left(\frac{Y}{2} \right)$ and $C' =  \sqrt{\frac{3}{2}} \left(\frac{B-L}{2} \right)$. Therefore the equation above becomes:
\begin{equation}
Y' = \sqrt{\frac{3}{5}} T_{3R}+ \sqrt{\frac{2}{5}} C' \,.
\end{equation}
Finally, when you use this expression with the coupling constants (squared) and do the matching, you get the relation which appears in your question.
(note that: $SU(4)_C \to SU(3)_C\times U(1)_{B-L}$ .)
