Let $E/\overline{\mathbb{F}}_p$ be a supersingular elliptic curve and $\mathcal{O}=\operatorname{End}E$. Let $\ell\neq p$ be a prime. Voi21, Theorem 42.1.9 showed that $E[\ell]\cong \mathbb{Z}/\ell\mathbb{Z} \otimes \mathbb{Z}/\ell\mathbb{Z}$ as abelian groups, and the endomorphism ring of this abelian group is $\mathcal{O}/\ell\mathcal{O}\cong\operatorname{End}E[\ell] \cong M_2(\mathbb{Z}/\ell\mathbb{Z})$.
Can we view these as ring isomorphisms? Given endomorphisms $\phi, \psi$, their restriction $\phi_\ell, \psi_\ell$ on $E[\ell]$ are in $\text{End}E[n]$ and after fixing a basis $(P_1, P_2)$ for $E[\ell]$, let $\phi_\ell(P_1)=aP_1+bP_2$, $\phi_\ell(P_2)=cP_1+dP_2$, $\psi_\ell(P_1)=xP_1+yP_2$, and $\psi_\ell(P_2)=zP_1+wP_2$ then their actions on $E[\ell]$ have matrix representations in $M_2(\mathbb{F}_\ell)$
$$M_\phi=\begin{pmatrix}a&b \\\ c&d\end{pmatrix}, M_\psi=\begin{pmatrix}x & y \\\ z & u \end{pmatrix}$$
Further, if $\lambda = \psi\circ \phi$ then $\lambda_\ell(P_1)= axP_1 + bzP_1 + ayP_2+buP_2$ and $\lambda_\ell(P_2)= cxP_1+dzP_1 + cyP_2+duP_2 $, and $M_\lambda = M_\phi M_\psi$. If we define multiplication $\phi_\ell \cdot \psi_\ell:=\lambda_\ell$ in $\operatorname{End}E[\ell]$, then is $\mathcal{O}/\ell\mathcal{O} \cong M_2(\mathbb{F}_\ell)$ a ring isomorphism?