This is the vector bundle operations defined in Lang's Fundamental of Differential Geometry which I do not grasp the essence here.
Consider subcategories of banach spaces $\mathcal{A,B,C}$ and let $\lambda:\mathcal{A\times B}\to \mathcal{C}$ be a functor which is contravariant in the first and covariant in the second.
Let $f:E'\to E$ and $g:F\to F'$ be two continuous linear maps between topological vector spaces with $f\in Mor_\mathcal{A}(E',E),g\in Mor_{\mathcal{B}}(F,F')$, then by definition, we have $L(E,E')\times L(F,F')\to L(\lambda(E,F),\lambda (E',F'))$ assigning $(f,g)$ to $\lambda(f,g)$.
Q1: What is this definition talking about here? Can some one give me a concrete non-trivial example other than tangent bundle or cotangent bundle or symmetric forms or alternating forms(these are easy to understand) say both $\mathcal{A}$ and $\mathcal{B}$ position non-trivial?
Q2: What is the reason for this definition to be reasonable to first guess?