A twisted bundle map have a reversed base morphism; we have $(f’,f): E \rightarrow F$ where $f’: E \rightarrow F$ and $f: M \leftarrow N$.

We see that the identity bundle map has a twisted avatar by reversing the base map, which we can do because the map is invertible.

They form a category, since a composition of twisted maps is twisted; and there are twisted identity maps.

The natural example of the twisted category is the dual category of the tangent category. Recall a natural bundle is a transformation. Say $\tau: T \rightarrow 1$; and the tangent bundle is the standard example of this.

Hence, we can define a twisted transformation as $\tau: T \rightarrow S$ where T is a functor and S is a cofunctor.

Now given a natural bundle $\tau: T \rightarrow 1$, whose functor $T$ takes values in vector space; we can define a twisted version by taking the dual, and we get $\tau^*: T^* \rightarrow 1$.