Let $\delta_X: \ker c^X \to \operatorname{coker} a^X$ denote the connecting homomorphism in the [snake lemma](/theorems/1930) exact sequence associated to $X$:
Then $F$ induces $R$-linear maps between the corresponding kernel and cokernel terms, and these maps form a morphism of [snake lemma](/theorems/4533) exact sequences. In particular, the connecting homomorphisms are natural: