Remark 1

Let V and $V'$ be B-modules. The adjunction isomorphism

$\gamma :$ Hom$_A(\mathscr{T}(V),\mathscr{T}(V'))\longrightarrow$ Hom$_B(V,\mathscr{H}(\mathscr{T}(V')))$

satisfies the relation $\gamma (\mathscr{T}(f)) =\beta_{V'}\circ f$ for every $f\in$ Hom$_B(V,V')$ because

Let M and $M'$ be A-modules; the inverse of the adjunction isomorphism

$\gamma^{-1}:$ Hom$_B(\mathscr{H}(M),\mathscr{H}(M'))\longrightarrow$ Hom$_A(\mathscr{T}(\mathscr{H}(M)),M')$

satisfies the relation $\gamma^{-1}(\mathscr{H}(u)) =u\circ \alpha_M$ for every $u\in$ Hom$_B(M,M')$. Indeed, we have the relations

for all $u\in$ Hom$_B(M,M'),v\in \mathscr{H}$ (M), and $p\in P$.