# Naturality of tensors in Differential Geometry

I have always had a hard time understanding the big picture of tensors and tensor fields. I have no problem understanding why low type tensors and tensor fields such as \begin{align} \text{$scalars$ and $smooth$ $functions$ --- type $(0,0)$,}\\ \text{$vectors$ and $vector$ $fields$ --- type $(1,0)$,}\\ \text{$covectors$ and $differential$ $forms$ --- type $(0,1)$,}\\ \text{$linear$ $transformations$ and $vector$-$field$ $morphisms$ --- type $(1,1)$,}\\ \text{$inner$ $products$ and $Riemannian$ $metrics$ --- type $(0,2)$ } \end{align} are so useful and natural. As these objects all share a lot of algebraic structure I understand the reason to encapsulate them within the notion of tensor, in an algebraic context. But, from an Analysis-Geometry setting, they are way different objects, so I see no natural reason to join together all these objects and expect the resulting object (tensors) to be of so much use in Differential Geometry. Yet they are, and they are everywhere!!

Q: Am I missing some reason of Analytic-Geometric character which motivates this generalization? If not, how does it turn out that an object whose generalization seem to be natural only algebraically end up playing such a central role in Differential Geometry? Maybe I am just underestimating the role that the algebraic structure play in this context?