  In this article, we construct zeta morphisms for the universal deformations
of odd absolutely irreducible two dimensional mod p Galois representations
satisfying some mild assumptions, and prove that our zeta morphisms interpolate
Kato's zeta morphisms for Galois representations associated to Hecke eigen cusp
newforms. The existence of such morphisms was predicted by Kato's generalized
Iwasawa main conjecture. Based on Kato's original construction, we construct
our zeta morphisms using many deep results in the theory of p-adic (local and
global) Langlands correspondence for GL_{2/Q}. As an application of our zeta
morphisms and the resent article {KLP19}, we prove a theorem which roughly
states that, under some mu=0 assumption, Iwasawa main conjecture without p-adic
L-function for f holds if this conjecture holds for one g which is congruent to
f.
