Wedge product/Universal property/Fact/Proof

We use the notation from fact. The assignment

${\displaystyle e_{(v_{1},\ldots ,v_{n})}\longmapsto \psi (v_{1},\ldots ,v_{n})}$

defines, due to fact, a ${\displaystyle {}K}$-linear mapping

${\displaystyle {\bar {\psi }}\colon H\longrightarrow W.}$

As ${\displaystyle {}\psi }$ is multilinear and alternating, ${\displaystyle {}{\bar {\psi }}}$ sends the linear subspace ${\displaystyle {}U\subseteq H}$ to ${\displaystyle {}0}$. According to fact, there exists a ${\displaystyle {}K}$-linear mapping

${\displaystyle {\tilde {\psi }}\colon H/U\longrightarrow W,}$

which is compatible with ${\displaystyle {}{\bar {\psi }}}$.
The uniqueness follows from the fact that the ${\displaystyle {}v_{1}\wedge \ldots \wedge v_{n}}$ form a generating system of ${\displaystyle {}\bigwedge ^{n}V}$, and these have to be sent to ${\displaystyle {}\psi (v_{1},\ldots ,v_{n})}$.