I am reading Rigidity of Gradient Ricci Solitons by Petersen and Wylie. In it is the following:
Consider the exterior covariant derivative $$ \mathrm{d}^\nabla : \Omega^p(M, TM) \to \Omega^{p+1}(M, TM) $$ for forms with values in tangent bundle. The curvature can then be reinterpreted as the 2-form $$ R(X,Y)Z = ( (\mathrm{d}^\nabla \circ \mathrm{d}^\nabla)(Z) ) (X,Y) .$$
I do not understand the last sentence. For context, I understand the operators, not the notation. For me, the curvature operator as written above is a (1,3)- tensor and the exterior covariant derivative applies to forms. My best guess is that we can consider $Z$ as a (1,0)-tensor and treat it as a 0-form (or just a function). In this case, $ (\mathrm{d}^\nabla \circ \mathrm{d}^\nabla)(Z)$ would then be a (1,2) tensor and assuming it is somehow tensorial over $Z$, then at least the tensor description matches. However this requires making sense of $ \mathrm{d}^\nabla Z$ as a (1,1)-tensor and I cannot make sense of it.
Any help would be highly appreciated.
