An $S$-graded set is an $S$-indexed family of sets$\{X_s\}_{s\in S}$. This can equivalently be described as a function$S \to Set$, or as a function $X \to S$ (a bundle with $X_s$ the fiber over $s\in S$).

The elements of $X_s$ are often said to have degree$s$.

Given a pair of $S$-graded sets $\{X_s\}_{s\in S}$ and $\{Y_s\}_{s\in S}$, a homomorphism between them is an $S$-indexed family of functions $\{f_s \colon X_s \to Y_s \}_{s\in S}$. This can equivalently be described as a natural transformation between the two associated functions $S \to Set$, or as a function from $X$ to $Y$ that make the diagram with the associated functions $X \to S$ and $Y \to S$commute.