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I have defined two CTensor (h and H),

h = CTensor[{{1 + ϵ ψ[], a[] ϵ VarD[x[]][α[]], a[]  ϵ VarD[y[]][α[]], a[] ϵ  VarD[z[]][α[]]},
             {ϵ VarD[x[]][α[]], a[] (1 - ϵ ϕ[]), a[] ϵ β1[], a[] ϵ β2[]},
             {ϵ VarD[y[]][α[]], -a[] ϵ β1[], a[] (1 - ϵ ϕ[]), a[] ϵ β3[]},
             {ϵ VarD[z[]][α[]], -a[] ϵ β2[], -a[] ϵ \ β3[], a[] (1 - ϵ ϕ[])}}, {-ℬ, ℬ}];

and

H = CTensor[{{1 - ϵ ψ[], -ϵ VarD[x[]][α[]], -ϵ VarD[y[]][α[]], -ϵ VarD[z[]][α[]]},
             {-ϵ VarD[x[]][α[]]/a[], (1 + ϵ ϕ[])/ a[], ϵ β1[]/a[], ϵ β2[]/a[]},
             {-ϵ VarD[y[]][α[]]/a[], -ϵ β1[]/ a[], (1 + ϵ ϕ[])/a[], ϵ β3[]/ a[]},
             {-ϵ VarD[z[]][α[]]/a[], - ϵ β2[]/ a[], -ϵ β3[]/a[], (1 + ϵ ϕ[])/ a[]}}, {-ℬ, ℬ}];

then using these CTensors and their covariant derivatives I have defined a new tensor. But the final it can not calculate the their components.

DefTensor[T[γ, -δ, -ζ], M];
IndexSet[T[γ_, -δ_, -ζ_], (H[-σ, γ]* CD[-δ]@h[-ζ, σ] -  H[-σ, γ]*CD[-ζ] @ h[-δ, σ])];

But when I use

T[{1, ℬ}, {1, -ℬ}, {2, -ℬ}]

I get

T^1_12
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The pattern T[γ_, -δ_, -ζ_] only accepts abstract indices in the second and third indices, because something like {1, -ℬ} does not match a pattern like -δ_. Hence use something like

T[γ, -δ, -ζ] Basis[-γ, {1, ℬ}] Basis[{1, -ℬ}, δ] Basis[{2, -ℬ}, ζ] // ContractBasis
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