This may be a dumb question.

$\mathcal{M}(r,d)$ is a coarse moduli scheme for semistable pairs $(E,\phi:E \rightarrow K_X \otimes E)$ of rank $r$, degree $d$ on a smooth projective curve $X$ over $\mathbb{C}$.
Then,the Hitchin morphism from $\mathcal{M}(r,d)$ to $H^0(X,K_X) \times ...\times H^0(X,K_X^{\otimes r})$ is defined a pair $(E,\phi)$ to its characteristic polynomial. Explicitly, $(E,\phi) \mapsto (\operatorname{Tr}(\wedge^1\phi),...,\operatorname{Tr}(\wedge^r\phi))$
But strictly, this is not a scheme morphism in the sense of Hartshorne's algebraic geometry.

My idea:
Let $U_i=Spec(B_i)\subset{\mathcal{M}(r,d)}$ be affine cover. First, define $Sym(V^*)\rightarrow B_i$.
If it is possible to glue them, we get the morphism $\mathcal{M}(r,d) \rightarrow Spec(Sym(V^*))$ $(V:=H^0(X,K_X) \times ...\times H^0(X,K_X^{\otimes r}))$, using $Hom_{sch}(X,Spec(R))\simeq Hom_{ring}(R,\Gamma(X,O_X))$
But can we gule them? Or this is a completely wrong direction?