Production and backreaction of fermions from axion- SU(2) gauge fields during inflation

$SU(2)$ gauge fields and axions can have a stable, isotropic and homogeneous configuration during inflation. However, couplings to other matter species lead to particle production, which in turn induces backreaction on and destabilization of the non-Abelian and axion background. In this paper, we first study the particle production by a $SU(2)$ gauge field coupled to a massive Dirac doublet. To carry out this calculation we have made two technical improvements compared to what has been done in the literature. First, we apply the antisymmetrization of the operators to treat particles and antiparticles on equal footing; second, to deal with the UV divergences, we apply instantaneous subtraction. We find that the backreaction of produced fermions on the $SU(2)$ background is negligible for model parameters of observational interest. Next, we consider production of fermions due to coupling to the axion. The tree-level backreaction on the gauge fields, as well as on the axion, is vanishingly small. We also provide an estimate for the loop effects.