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 $i=1,\mathrm{.}\mathrm{.}\mathrm{.},k$. In this short note, we obtain necessary and sufficient conditions for the consistency of the system of $\star$-Sylvester equations $A_{i}X-X^{\star }{B}_i=C_i$, for $i=1,\mathrm{.}\mathrm{.}\mathrm{.},k$, where $\star$ denotes either the transpose or, when $\mathbb{F}=\mathbb{C}$, the conjugate transpose.