A note on the consistency of a system of star-Sylvester equations

Let ${\mathbb F}$ be a field with characteristic not $2$, and $A_i\in{\mathbb F}^{m\times n},B_i\in{\mathbb F}^{n\times m},C_i\in{\mathbb F}^{m\times m}$, for $i=1,...,k$. In this short note, we obtain necessary and sufficient conditions for the consistency of the system of $\star$-Sylvester equations $A_iX-X^\star B_i=C_i$, for $i=1,...,k$, where $\star$ denotes either the transpose or, when ${\mathbb F}={\mathbb C}$, the conjugate transpose.