Let $X$ be a set and let $\text{Mo}(X)=\bigcup_{n\in\mathbb{N}} X^{[1,n]}$. Then $\text{Mo}(X)$ together with the law $(w,w')\mapsto ww'$, where $ww'$ denotes the juxtaposition of the sequences $w$ and $w'$, is a monoid. The identity element of $\text{Mo}(X)$ is the unique empty sequence $e\in X^{[1,0]}$.
We have the following convention:
Let $M$ be a monoid with a multiplicative law. If $(x_\alpha)_{\alpha\in\emptyset}$ is the empty family of elements of $M$, then $\prod_{\alpha\in\emptyset} x_\alpha=e$.
We also have the following result:
If $w=(x_i)_{1\leq i\leq n}$ for $x\in X^{[1,n]}$ and $n\in\mathbb{N}$, then $w=\prod_{i=1}^nx_i$.
Now, let $w=(x_i)_{1\leq i\leq 0}$ for $x\in X^{[1,0]}$: i.e. $w=e$. By our convention, the composition of the family $(y_\alpha)_{\alpha\in\emptyset}$ of elements of $\text{Mo}(X)$ is equal to $e$: that is, $$e=\prod_{\alpha\in\emptyset}y_\alpha$$ for $y\in\text{Mo}(X)^{\emptyset}$. But $x\in X^\emptyset$ and not in $\text{Mo}(X)^{\emptyset}$, so it cannot be that $e=\prod_{i=1}^0x_i$.
How can I resolve this? Should I say that $X^\emptyset=\text{Mo}(X)^\emptyset$ (but this seems dubious)?