of $\mathcal{E}$ is less than three; if $X$ is isomorphic to a hyperquadric $\mathbb{Q}^n$ ($n\geqq 3$) and $c_1(\mathcal{E})< 2$. (Here, for simplicity as in [@pswnef], we identify the first Chern class $c_1(\mathcal{E})$ with the corresponding non-negative integer via the isomorphism ${\mathop{\rm Pic}\nolimits}X\cong \mathbb{Z}$ by abuse of notation.) Thus the classification [@pswnef] of the pairs $(X,\mathcal{E})$ in the above cases has fundamental importance in their proof. Towards the classification of “Mukai bundles”, it seems therefore natural to consider the classification of $(X,\mathcal{E})$ in the next cases, e.g., $X=\mathbb{P}^n$ and $c_1(\mathcal{E})=3$, or $X=\mathbb{Q}^n$ and $c_1(\mathcal{E})=2$. However the classification of $(X,\mathcal{E})$ with $\mathcal{E}$ nef of an arbitrary rank $r$ in the next cases has not been pursued over twenty years.

One of the reason why such research has not been pursued seems to come from the fact that it is uncertain how to describe nef bundles in general. In order to overcome this situation, in this paper, we first propose a framework to describe nef bundles on a projective space or a smooth hyperquadric. Following this framework, we secondly give a new proof of the above classification of $(X,\mathcal{E})$ due to Peternell-Szurek-Wiśniewski [@pswnef]. In their proof, Peternell-Szurek-Wiśniewski analyze the contraction morphism of an 