I wondered how it works exactly to induce a $\mathrm{spin^c}$-structure if a spin structure is given. I wanted to use the following definitions as used in Friedrich`s "Dirac operators in Riemannian Geometry"
Let $(Q,\pi_Q,M;\operatorname{SO}(n))$ be a $\operatorname{SO}(n)$-principle bundle on a manifold $M$. The maps $\lambda:\operatorname{Spin}(n)\to \operatorname{SO}(n)$ and $\lambda_\mathrm{c}:\mathrm{Spin^c}(n)\to \operatorname{SO}(n)$ below are the respective standard covering maps. The horizontal arrows are the actions of the groups on the fibers.
Definition A spin structure on $Q$ is a pair $(P,\varLambda)$ consisting of a $\operatorname{Spin}(n)$-principal bundle $(P,\pi_P,M;\operatorname{Spin}(n))$ such that $\varLambda:P\to Q$ is a $2$-fold covering bundle map for which the following diagram commutes
Definition A $spin^c$ structure on $M$ is a pair $ (P^{\mathrm{c}},\varLambda_{\mathrm{c}})$ consisting of a $\mathrm{Spin^c}(n)$-principle bundle $(P^{\mathrm{c}},\pi_P,M;\mathrm{Spin^c}(n))$ over $M$ and a fiber map $\varLambda_{\mathrm{c}}:P\to Q$ such that the following diagram commutes:
Now a typical assertion is that each spin structure induces a canonical $\mathrm{spin^c}$ structure. I tried to show that via connecting the commutative diagrams, and using the fact that there is a natural inclusion of the spin group in the $\mathrm{spin^c}$ group. That means I get a diagram, where the "upper" and "big" part commutes, but I do not know how to define $\varLambda_{\mathrm{c}}$:
The map $\Lambda_c: P \times _{Spin(n)} Spin^{c}(n) \rightarrow Q$ is given by $\Lambda_{c}([p,c])=\Lambda(p)\lambda_{c}(c).$