When I was learning differential geometry, I was told that differential of a function can be viewed as a functor, i.e. sending manifold into tangent space and function into its differential. Unfortunately, I believe that it's simply abstract nonsense, since I cannot find any references on the properties of tangent functor, for example, when it's exact, when it keeps injective (a bit wired, since there are piles of injection, immersed or embedded for example), or whether it keeps the limit or colimit. (for finite product, it's evident)
So is there any reference for the properties of tangent functor? Thanks!