Remark

If the left adjoint $u^*$ of $u : E \to E'$ is everywhere defined, and if $s_{\Phi'}$ and $d_{\Phi'}$ are injective, formula (30) shows that $u$ is the right adjoint of $u^*$.

It follows from (30) that, if $u_1$ and $u_2$ are two homomorphisms from $E$ into $E'$ admitting everywhere defined adjoints, and if $c$ is an element of the center of $A$, one has

Moreover, if $E''$ is a third $A$-left module, $F''$ a third $A$-right module (resp. left module), $\Phi''$ a bilinear form (resp. sesquilinear for $J$) on $E'' \times F''$, and if $u'$ is a homomorphism from $E'$ into $E''$ admitting an everywhere defined (left) adjoint, one has

In particular, if $u$ is an isomorphism of $E$ onto $E'$, and if the adjoints $u^*$ and $(u^{-1})^*$ are everywhere defined, $u^*$ is an isomorphism of $F'$ onto $F$, and one has $(u^*)^{-1} = (u^{-1})^*$. Analogous properties for right adjoints.