In Huybrechts' book "Fourier Mukai transforms in algebraic geometry" he defines (def 1.39) an exact (or triangulated) functor between triangulated categories as follows. An exact functor is an additive functor $F: D \rightarrow D'$ such that
i) There exists a natural isomorphism $F \circ T_{D} \simeq T_{D'} \circ F$,
ii) Any distinguished triangle $$ A \rightarrow B \rightarrow C \rightarrow A[1] $$ is mapped to a distinguished triangle $$ F(A) \rightarrow F(B) \rightarrow F(C) \rightarrow F(A)[1] $$ where $F(A[1])$ is identified with $F(A)[1]$ via the previous isomorphism.
I don't really understand this definition. Does it mean that there exists a natural isomorphism as in i) such that ii) holds, or that for every isomorphism such in i) it is true that distinguished triangles are mapped into distinguished triangles?
I tried to prove that by changing the natural isomorphism in i) one gets isomorphic triangles $$ F(A) \rightarrow F(B) \rightarrow F(C) \rightarrow F(A)[1] $$ but I am not able to define the isomorphism of triangles.