# $\frac{\partial C_{BS}}{\partial T}$ in local volatility derivation in terms of implied volatility

In Gatheral's book, in the derivation of local volatility in terms of implied volatility, we use the regular Dupire formula

$$\frac{\partial C}{\partial T} = \frac{1}{2} \sigma^{2}K^{2}\frac{\partial^{2}C}{\partial^{2}K} + (r_{T}-q_{T}) \left ( C - K \frac{\partial C}{\partial K} \right ) - r_{T}C$$

and reparametrize with $$y = \ln \frac{K}{F_{T}}$$ and $$w = \Sigma(K,T)^{2}T$$ where $$\Sigma(K,T)$$ is the implied volatility and get:

$$\frac{\partial C}{\partial T} = \frac{v_{L}}{2} \left [ \frac{\partial^{2}C}{\partial^{2}y} - \frac{\partial C}{\partial y} \right ] + \mu_{T}C$$

where $$v_{L}$$ is the local variance.

Next, we set the market price equal to the B-S price to write the formula above in terms of derivatives of the B-S function:

$$C(S_{0}, K, T) = C_{BS}(S_{0}, K, \Sigma(K,T), T)$$

So far so good.

But in the next step when we write $$\frac{\partial C}{\partial T}$$ in terms of the B-S call function, we get:

$$\frac{\partial C}{\partial T} = \frac{\partial C_{BS}}{\partial T} + \frac{\partial C_{BS}}{\partial w}\frac{\partial w}{\partial T}$$

However, I would expect it to be

$$\frac{\partial C}{\partial T} = \frac{\partial C_{BS}}{\partial T} + \frac{\partial C_{BS}}{\partial w}\frac{\partial w}{\partial T} + \frac{\partial C_{BS}}{\partial y}\frac{\partial y}{\partial T}$$

since a shift in maturity implies a shift in the forward and thus in the log-moneyness $$y$$. Gatheral's explanation is that it "follows from the fact that the only explicit dependence of the option price on T [...] is through the forward price". Why do we not also take into account the log-moneyness shift here?