Looks like this may have been related to Hw2 Problem II.1, explicitly write our the U_L matrix.

ClearAll[Phi, PhiPrime, phi1, phi2, w1, w2, w3, pretty, uL, sigma0]
$Assumptions = w1 > 0 && w2 > 0 && w3 > 0; (* i.e. real, don't actually care about the sign. *)
w = {w1, w2, w3};
Phi = {phi1, phi2};
pretty = {Conjugate[phi1] -> (SuperStar[(Subscript[ϕ, 1])]),Conjugate[phi2] -> (SuperStar[(Subscript[ϕ, 2])]),phi1 -> (Subscript[ϕ, 1]),phi2 -> (Subscript[ϕ, 2]), w1 -> (Subscript[ω, 1]), w2 -> (Subscript[ω, 2]), w3 -> (Subscript[ω, 3])};
sigma0 = IdentityMatrix[2];
uL = sigma0 + ((1/2)(I PauliMatrix[#] w[[#]] &/@ Range[3]) // Total);
uLdagger =  sigma0 + ((1/2)(-I ConjugateTranspose[PauliMatrix[#]] w[[#]] &/@ Range[3]) // Total);
PhiPrime = uL . Phi;

deltaPhi = (PhiPrime - Phi) // Simplify;
(*deltaPhi*)
deltaPhiConj = deltaPhi// Conjugate // ComplexExpand;
Grid[{
{"Φ = ", Phi  /. pretty// MatrixForm },
{"Φ' = ", PhiPrime  /. pretty// MatrixForm },
{"(Subscript[U, L]) = ", uL  /. pretty// MatrixForm },
{"(SuperDagger[(Subscript[U, L])]) = ", uLdagger  /. pretty// MatrixForm },
{"δ (SuperDagger[Φ]) = ",deltaPhiConj /. pretty// MatrixForm},
{"δ Φ = ",deltaPhi/. pretty// MatrixForm }
}]

((Phi // Conjugate) . deltaPhi + ( deltaPhiConj) . Phi // Expand // Simplify )/. pretty

(PauliMatrix[#] // MatrixForm)&/@ Range[3]

(PauliMatrix[#] // ConjugateTranspose// MatrixForm)&/@ Range[3]
((PauliMatrix[#] - (PauliMatrix[#] // ConjugateTranspose))// MatrixForm)&/@ Range[3]