Say we define for some arbitrary $S$ a family of functions $f_i:X_i\to S$, for $$X_i=\prod _{Y\in \mathcal Y(i)}Y$$
Now assume that $\mathcal Y(1)=\emptyset$. Then $X_1$ is a singleton set.
Is there a standard notation for how to evaluate $f_1$? On the one hand, I'd be tempted to write $f_1()=s$, since the cartesian product is empty, so that we should input "an empty set of parameters". On the other hand, $f_1()$ doesn't seem to make much sense as notation, and I'd be tempted to write $f_1(\{\})=s$ or something. But this would make the syntax for evaluating $f_1$ very different from $f_j$ for $\mathcal Y(j)\neq \emptyset$.
Is there a standard way of writing the evaluation of $f_1$ here?
If $\mathcal Y(1)=\emptyset$, then $X_1=\{\emptyset\to \emptyset\}=\emptyset^\emptyset$, so the domain of $f_1$ is precisely $\{\emptyset\to \emptyset\}$. You can say that $t$ is the only function that maps the empty set into itself, and then $f(t)=s$, or simply $$f(\emptyset\to \emptyset)=s$$
I doubt that there is a standard notation for this.
Given that a function $\emptyset\to\emptyset$ is a subset of $\emptyset\times\emptyset$, and $\emptyset\times\emptyset=\emptyset$, it is clear that $$\emptyset\to\emptyset=\emptyset$$ so you can also write that $X_1=\{\emptyset\}=\{0\}=1$.