Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 207 / 4 / a

$N(t)$ is the event counting process, while $Y(t)$ is the at-risk process. Their key relationship is that, conditionally on the observed past, the expected event increment is $Y(t)dH(t)$. It separates exposure to risk from event occurrence and naturally incorporates right censoring.

Before the observed endpoint, $Y+N=1$. If censoring occurs before the event, both processes are zero after the censoring time, so their sum is zero. With the inclusive endpoint conventions in the question, at an observed event time $t=T\le C$ both are one, so the sum is two at that single instant.