Curvature of a superconnection

Question

Let $E\rightarrow X$ be a $\mathbb{Z}_2$-vector bundle (or superbundle for connoisseurs) and consider the superconnection $$A=\nabla + B$$ where $\nabla$ is a connection on $E$ and $B\in\Gamma(End(E))^-$ simply an odd endomorphism. I would like to understand how to compute the curvature $A^2$: I can of course write $$A^2=\nabla^2 + \nabla B + B\nabla + B^2$$ but I would like to understand whether it makes sense to write $$A^2(X,Y)\in \Gamma(End(E))$$ just like for connections. The problem is that $A^2$ is the sum of a 2-form, a 1-form and a simple endomorphism. Hence it makes no sense a priori to write $A^2(X,Y)$. Nevertheless I wonder based on the fact one can always write $$A_XA_Ys := (\nabla_X+B)(\nabla_Y+B)s$$ whether it's possible to make a canonical sense of $A^2(X,Y)$.