**Background**

For definiteness (even though this is a categorical question!) let's agree that a *vector space* is a finite-dimensional real vector space and that an *associative algebra* is a finite-dimensional real unital associative algebra.

Let $V$ be a vector space with a nondegenerate symmetric bilinear form $B$ and let $Q(x) = B(x,x)$ be the associated quadratic form. Let's call the pair $(V,Q)$ a **quadratic vector space**.

Let $A$ be an associative algebra and let's say that a linear map $\phi:V \to A$ is **Clifford** if
$$\phi(x)^2 = - Q(x) 1_A,$$
where $1_A$ is the unit in $A$.

One way to define the Clifford algebra associated to $(V,Q)$ is to say that it is universal for Clifford maps from $(V,Q)$. Categorically, one defines a category whose objects are pairs $(\phi,A)$ consisting of an associative algebra $A$ and a Clifford map $\phi: V \to A$ and whose arrows
$$h:(\phi,A)\to (\phi',A')$$
are morphisms $h: A \to A'$ of associative algebras such that the obvious triangle commutes:
$$h \circ \phi = \phi'.$$
Then the **Clifford algebra of $(V,Q)$** is the universal initial object in this category. In other words, it is a pair $(i,Cl(V,Q))$ where $Cl(V,Q)$ is an associative algebra and $i:V \to Cl(V,Q)$ is a Clifford map, such that for every Clifford map $\phi:V \to A$, there is a unique morphism
$$\Phi: Cl(V,Q) \to A$$
extending $\phi$; that is, such that $\Phi \circ i = \phi$.

(This is the usual definition one can find, say, in the nLab.)

**Question**

I would like to view the construction of the Clifford algebra as a functor from the category of quadratic vector spaces to the category of associative algebras. The universal property says that if $(V,Q)$ is a quadratic vector space and $A$ is an associative algebra, then there is a bijection of hom-sets

$$\mathrm{hom}_{\mathbf{Assoc}}(Cl(V,Q), A) \cong \mathrm{cl-hom}(V,A)$$

where the left-hand side are the associative algebra morphisms and the right-hand side are the Clifford morphisms.

My question is whether I can view $Cl$ as an adjoint functor in some way. In other words, is there some category $\mathbf{C}$ such that the right-side is $$\mathrm{hom}_{\mathbf{C}}((V,Q), F(A))$$ for some functor $F$ from associative algebras to $\mathbf{C}$. Naively I'd say $\mathbf{C}$ ought to be the category of quadratic vector spaces, but I cannot think of a suitable $F$.

I apologise if this question is a little vague. I'm not a very categorical person, but I'm preparing some notes for a graduate course on spin geometry next semester and the question arose in my mind.