The mirror map and other natural tangent tensors

We review the intrinsic geometry of the tangent bundle of a differentiable manifold $M$, aside from any non-natural structures. We recall the properties of the mirror map $B\in \mathrm{E}\mathrm{n}\mathrm{d}(TTM)$, known also as the canonical endomorphism or the almost-tangent structure, solve some cohomological questions and raise other induced by the ${\mathrm{d}}_B$ operator.