I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on triangulated category. I found there was also a definition of "frobenius morphism" the definition is as follows:

There are two categories C and D. f_*:D--->C, f^* :C--->D is left adjoint to f_*, we call f_* is a Frobenious morphism if there exists an auto-equivalence G of C such that composition f^* G is right adjoint to f_*.

First question is:is there any relationship between this two frobenius morphism?

Second question is:does frodenius category play roles in algebraic geometry?

All the comments related to this are welcomed.