Comparison of types for inner forms of ${\mathrm{G}\mathrm{L}}_N$

Let $F$ be a non-archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. To construct types for supercuspidal representations of $G$, simple types by S'echerre–Stevens and Yu’s construction are already known. In this paper, we compare these constructions. In particular, we show essentially tame supercuspidal representations of $G$ defined by Bushnell–Henniart are nothing but tame supercuspidal representations defined by Yu.