volatility smile - In Dupire's paper, why is $(S_t, t)$ in the $(K, T)$ space?

I'm new to local volatility model.

From Dupire's paper and most of the textbooks, they derived the local volatility $\sigma(K, T)$ in the $(K, T)$ (i.e., strike and maturity) space, from call prices or the implied volatility surface.

However, by definition, local volatility is a function in terms of $(S_t, t)$, i.e., instantaneous underlying price and time.

How to relate these two?