I don't fully understand what a coordinate process is and why they are used. My lecture notes say the following:
For an arbitrary measurable space $(S,\mathcal{S})$ we denote by $S^{[0,\infty)}$ the space of all functions $y: [0,\infty)\to S,t\mapsto y(t)$. We introduce the coordinate process $Y = (Y_t)_{t\geq 0}$ on $S^{[0,\infty)}$ by setting $Y_{t}(y):=y(t)$ for $y\in S^{[0,\infty)}$ and $t\geq 0$.
So we have a random process $Y$ on the spaces of all functions on $S$ and the $t$-position for some function $y$ of this process is equal to $y(t)$. What exactly is the point here, why do we need to introduce this? Does this mean when we show properties for this process $Y$, we automatically show them for any other stochastic process, such as RCLL processes?
Also my notes talk about $Y$ being the canonical $\mathbb{P}_{\nu}$-Markov process (for some transition semigroup $(K_t)_{t\geq 0}$ and initial distribution $\nu$ on $(S,\mathcal{S})$). What exactly does 'canonical process' mean?