I was very suprised when I found this post since I am not aware of it although the Clifford algebra is my Phd subject. Indeed, the construction of Clifford algebras can be regarded as a left adjoint on a different category $\mathrm{HPL}^d(S)$, which also works for any forms of higher degrees with non-commutative coefficients. You can find this adjunction in Section 3.1 of my thesis https://drive.google.com/file/d/1MKS6Y1V1lEtt60C9ZARAD-UyVlxtr8X3/view?usp=drivesdk
The proposition contains the answer is Propostion 3.1.10. Reading through the above answers, I think the problem is that the notion of forms relies to much on a concrete presentation, as well as their coefficients are restricted to the commutative base ring. In fact, I believe that any algebra that is defined by some *averaging procedure* should be realized as a left adjoint by some modification of the construction of the category $\mathrm{HPL}^d(S).$