I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$.
Now, fiberwise it's all good. But I do not understand how to define on sections, as a map of sheaves of derivations.
More to the point, say we have a map of rings $\varphi: B \leftarrow A$ (which I am thinking of opposite to $f$ above, in other words $A$ is global functions on $N$ and $B$ is global functions on $M$). And say that $\varphi$ is over a ground ring $k$ (everything in site is commutative!).
Now, I do not see how to get a map $Der_k(B,B) \to Der_k(A,A) \otimes_A B$ (the latter being the pullback of $TN$ on $M$).
While there is an obvious map $Der_k(B,B) \to Der_k(A,B)$, given by restriction, (which can be used to prove functoriality of the cotangent bundle) I do not see why we should have that $Der_k(A,A)\otimes_A B = Der_k(A,B)$ (although of course it needn't even hold).
On the other hand one might define first the map on cotangent bundles and then declare this one as the transpose (but I'm trying to get grips with which one should be more "natural", in some very unfair sense).
I guess my last comment is important to me when $M$ and $N$ at the beginning are taken to be singular varieties instead of manifolds.