In what way is $\sigma(L) \in K(T^*X)$ in the $K$-theoretic formulation of the Atiyah-Singer index theorem?

Question

I am asking whether my interpretation of the $K$-theoretic description of the Atiyah-Singer index theorem is correct. First I state what I mean by "$K$-theoretic description".

Let $X$ be a manifold and $E, F \rightarrow X$ be a complex vector bundles. Denote by $T^*X \overset{\pi}{\rightarrow} X$ the canonical projection of the cotangent bundle. Consider an elliptic operator $L:C^\infty (E) \rightarrow C^\infty (F)$ of order $m$. The Atiyah-Singer index theorem states that $$ index_a (L) = \int_X \phi ( ch( \sigma_m(L)) \cup td(X), $$ where

• $index_a(L)=\dim Ker L- \dim CoKer L$ is the analytical index,
• $\phi : K(T^*M) \rightarrow K(X)$ means the Thom isomorphism,
• $ch : K(T^*X) \rightarrow H^*(T^*X, \mathbb{Q})$ means the Chern character, (it is defined for any manifold, not just $T^*X$)
• $\sigma _m (L) : \pi^*E \rightarrow \pi^*F$ denotes the symbol of $L$, defined by $$ \sigma_m {L}_{(x,v)} : E_x \rightarrow F_x \\ \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \;\; \; e \mapsto L \left( \frac{i^m}{m!} (g-g(x))^m \cdot f \right)(x), $$ where $x \in X$, $v \in T^*_xX$ (so one may consider $(x,v,e)$ as an element of $\pi^*E$), $g \in C^\infty(X)$ such that $dg(x)=v$, $f \in C^\infty (X,E)$ such that $f(x)=e$,
• $\cup:H^*(X) \times H^*(X) \rightarrow H^*(X)$ denotes the cup product,
• $td(X)$ denotes the Todd class of $X$.

Question:

Now, $ch$ is a map from the $K$-group $K(T^*X)$. In what way can $\sigma_m(L)$ be seen as an element in $K(T^*X)$?

Suspected answer:

We have the following characterization of $K$-groups: $K(X)=C(X) / \sim$, where $C(X)$ denotes the category which has as objects the vector bundle complexes with compact support over $X$ and as morphisms the usual vector bundle complex morphisms modulo homotopy and $E^\bullet \sim F^\bullet$ if exists a $G^\bullet \in C(X \times I)$ such that $i_0^* G \simeq E^\bullet$ and $i_1^* G \simeq F^\bullet$.

$L$ is elliptic, so by definition $\sigma _m(L)$ is an isomorphism, so $$ 0 \rightarrow \pi^* E \rightarrow \pi ^* F \rightarrow 0 $$ is a vector bundle complex over $T^*M$ with empty support, in particular the support is compact. In that way we can regard $\sigma_m(L)$ as an element in $K(T^*M)$.

The construction of the isomorphism $K(X) = C(X) / \sim$ is very involved. Is there an easier way to think of $\sigma_(L)$ as an element of $K(T^*M)$?