You could say "$f$ makes $X$ into a commutative $Y$-unary algebra" or that "$f$ is a commutative $Y$-unary algebra structure on $X$."

For instance, defining

$$f : \{N,E\} \times \mathbb{N}^2 \rightarrow \mathbb{N}^2$$ by $$f(N,(x,y)) = (x+1,y) \qquad f(E,(x,y)) = (x,y+1)$$

we have that $(\mathbb{N}^2,f)$ is the commutative $\{N,E\}$-unary algebra freely generated by $\{(0,0)\}$.

You could say "$f$ makes $X$ into a commutative $Y$-unary algebra" or that "$f$ is a commutative $Y$-unary algebra structure on $X$."

For instance, defining

$$f : \{N,E\} \times \mathbb{N}^2 \rightarrow \mathbb{N}^2$$ by $$f(N,(x,y)) = (x+1,y) \qquad f(E,(x,y)) = (x,y+1)$$

we have that $(\mathbb{N}^2,f)$ is the commutative $\{N,E\}$-unary algebra freely generated by $\{(0,0)\}$.