= Solution
In one standard <Schwinger-Keldysh propagator> convention, suppressing the momentum delta function,
$$
G_{RR}(k;\eta,\eta')
=\theta(\eta-\eta')f(\eta)f^*(\eta')
+\theta(\eta'-\eta)f^*(\eta)f(\eta'),
$$
$$
G_{LL}(k;\eta,\eta')
=\theta(\eta'-\eta)f(\eta)f^*(\eta')
+\theta(\eta-\eta')f^*(\eta)f(\eta'),
$$
$$
G_{RL}(k;\eta,\eta')=f^*(\eta)f(\eta'),
\qquad
G_{LR}(k;\eta,\eta')=f(\eta)f^*(\eta').
$$
The two <cosmological bulk-to-boundary propagator>[bulk-to-boundary propagators] to the late-time insertion are
$$
K_R(k,\eta)=f(k,0)f^*(k,\eta),
\qquad
K_L(k,\eta)=f^*(k,0)f(k,\eta).
$$
Interchanging every $L$ and $R$ gives the equivalent opposite contour-label convention. Direct differentiation of the supplied <Bunch-Davies vacuum> mode gives
$$
\boxed{f'(k,\eta)=\frac{Hk^2\eta}{\sqrt{2k^3}}e^{-ik\eta}
=\frac{H\sqrt k}{\sqrt2}\eta e^{-ik\eta}}.
$$
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