I was reading a textbook and I encountered this question. I was wondering why we don't consider $M^\dagger$ instead of $M^{T}$, so I didn't show this relation, could you please help me to show below relation?
Let $M: \mathcal{H}^{\tilde{A}} \rightarrow \mathcal{H}^{B}$ be a linear map and denote its transpose map by $M^{T}: \mathcal{H}^{\tilde{B}} \rightarrow \mathcal{H}^{A}$. Show that $$ I \otimes M\left|\phi_{+}^{A \tilde{A}}\right\rangle=M^{T} \otimes I\left|\phi_{+}^{\tilde{B} B}\right\rangle $$ where $\left|\phi_{+}^{\tilde{A} A}\right\rangle:=\sum_{y=1}^{|A|}|y y\rangle^{\tilde{A} A}$ and $\left|\phi_{+}^{\tilde{B} B}\right\rangle:=\sum_{y=1}^{|B|}|y y\rangle^{\tilde{B} B}$ and these are maximally entangled states.