Let $V$ be a vector space over a field $\mathbb{K}$ with characteristic zero and $Q$ be a quadratic form on $V$. Let $\mathcal{C}l(V,Q)$ be the associated Clifford algebra, with Clifford map $\varphi: V \to \mathcal{C}l(V,Q)$. By the universal property of Clifford algebras, if $\phi: V \to \mathcal{A}$ is another Clifford map, where $\mathcal{A}$ is an associative algebra with unit $e$, then there exists a unique homomorphism $h: \mathcal{C}l(V,Q) \to \mathcal{A}$ such that $h\circ \varphi = \phi$. Does it follow that $h(1) = e$ or not necessarily? I couldn't find an argument myself.
Note: By a Clifford map I mean a $\mathbb{K}$-linear map $\varphi: V \to \mathcal{F}$ where $F$ is some associative algebra with unit $1$ such that $\varphi(v)^{2} = Q(v)1$ holds for every $v \in V$.