\, ,$$ where ${{\mathbb{C}}}^4$ is the set of complex *row* 4-vectors, is constructed similarly and its fibres are the equivalence classes $[T,z^{\mathrm{t}}]_x^*$, $z^{\mathrm{t}}\in{{\mathbb{C}}}^4$ a row vector, such that $[\widetilde{T},\widetilde{z^{\mathrm{t}}}]_x^* =
[T,z^{\mathrm{t}}]_x^*$ if and only if $\widetilde{T}=R^{-1}_ST$ and $\widetilde{z^{\mathrm{t}}}=z^{\mathrm{t}}S^{-1}$ for some $S\in{\mathrm{Spin }}_0(1,3)$. Just as the sections of ${D{\mathcal{M}}}$ are called *spinors*, the sections of $D^*{\mathcal{M}}$ are called *cospinors*. We shall refer to *test spinors* as being the smooth and compactly supported sections of $D{\mathcal{M}}$, the space of which we denote $C^\infty_0(D{\mathcal{M}})$; *test cospinors* are similarly defined and are elements of $C^\infty_0(D^*{\mathcal{M}})$. As expected, there exists a natural pairing between spinors and cospinors: Set $v_x = [T,z_1^{\mathrm{t}}]_x^*$ and $u_x\in [T,z_2]_x$, then $v_x(u_x) = z_1^{\mathrm{t}}z_2$ is a scalar.

We are now in a position to define the Dirac adjoint operation ${\,}^+ :
D{\mathcal{M}}\mapsto D^*{\mathcal{M}}$ which is given by $$[T,z]^+_x = [T,z^\dagger \gamma_0]^*_x \, .$$

Any local section $E:{\mathcal{M}}\mapsto S{\mathcal{M}}$ of $S{\mathcal{M}}$ determines a local frame $e^a_\alpha$ by $\varphi\circ E$ and local sections $E_A$ of $D{\mathcal{M}}$, such that $E_A(x)=[E_x,z_A]$ where $\{ z_A \}_{A=0,1,2,3}$ is the canonical basis of ${\mathbb{C}}^4$. The dual frames $e_a^\alpha$, $E^A$ are defined through $e^\alpha \cdot e_\beta=\delta^\alpha_\beta$ and $E^A ( E_B)=\delta^A_B$. One may define a mixed tensor-spinor object $\gamma \in
