${\displaystyle {}\chi \in D_{2}^{\vee }}$

${\displaystyle D_{2}\longrightarrow K^{\times },}$

${\displaystyle \chi \circ \varphi \colon D_{1}\longrightarrow K^{\times }}$

${\displaystyle {}D_{1}^{\vee }}$

${\displaystyle {}\chi _{1},\chi _{2}\in D_{2}^{\vee }}$

${\displaystyle {}d\in D_{1}}$

${\displaystyle {}{\begin{aligned}((\chi _{1}\cdot \chi _{2})\circ \varphi )(d)&=(\chi _{1}\cdot \chi _{2})(\varphi (d))\\&=\chi _{1}(\varphi (d))\cdot \chi _{2}(\varphi (d))\\&=((\chi _{1}\circ \varphi )(d))\cdot ((\chi _{2}\circ \varphi )(d))\\&=((\chi _{1}\circ \varphi )\cdot (\chi _{2}\circ \varphi ))(d).\end{aligned}}}$

Also ist

${\displaystyle {}\varphi ^{\vee }(\chi _{1}\cdot \chi _{2})=(\chi _{1}\cdot \chi _{2})\circ \varphi =(\chi _{1}\circ \varphi )\cdot (\chi _{2}\circ \varphi )=\varphi ^{\vee }(\chi _{1})\cdot \varphi ^{\vee }(\chi _{2})\,}$

und die Zuordnung ist ein Gruppenhomomorphismus.

${\displaystyle {}\chi \in D_{3}^{\vee }}$

${\displaystyle {}(\psi \circ \varphi )^{\vee }(\chi )=\chi \circ (\psi \circ \varphi )=(\chi \circ \psi )\circ \varphi =\varphi ^{\vee }(\chi \circ \psi )=\varphi ^{\vee }(\psi ^{\vee }(\chi ))=(\varphi ^{\vee }\circ \psi ^{\vee })(\chi )\,.}$