YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
h(f(),a(),a()) -> h(g(),a(),a())
h(g(),a(),a()) -> h(f(),a(),a())
a() -> a'()
h(x,a'(),y) -> h(x,y,y)
g() -> f()
f() -> g()

CP(R):
h(f(),a'(),a()) -> h(g(),a(),a())
h(f(),a(),a'()) -> h(g(),a(),a())
h(g(),a'(),a()) -> h(f(),a(),a())
h(g(),a(),a'()) -> h(f(),a(),a())
h(f(),a(),a()) -> h(f(),a(),a())
h(g(),a(),a()) -> h(g(),a(),a())

critical-pair-closing system C:
h(f(),a(),a()) -> h(g(),a(),a())
h(g(),a(),a()) -> h(f(),a(),a())
a() -> a'()
g() -> f()
f() -> g()

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
h(f(),a(),a()) -> h(g(),a(),a())
h(g(),a(),a()) -> h(f(),a(),a())
a() -> a'()
g() -> f()
f() -> g()

CP(R):
h(f(),a'(),a()) -> h(g(),a(),a())
h(f(),a(),a'()) -> h(g(),a(),a())
h(g(),a'(),a()) -> h(f(),a(),a())
h(g(),a(),a'()) -> h(f(),a(),a())
h(f(),a(),a()) -> h(f(),a(),a())
h(g(),a(),a()) -> h(g(),a(),a())

critical-pair-closing system C:
a() -> a'()
g() -> f()

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a() -> a'()
g() -> f()

CP(R):


critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.