How is this derivation of a field transformation, in Weinberg's QFT book, performed?

I am reading Weinberg’s book Quantum theory of fields.

Could you explain to me the following things? Vol.1, page 60 (transcribed from this image):

To first order in $omega$ and $epsilon$, we have then
begin{align}
U(Lambda,a) left[tfrac12omega_{rhosigma}J^{rhosigma} -epsilon_rho P^rhoright] U^{-1}(Lambda,a)
& =
tfrac12(LambdaomegaLambda^{-1})_{munu} J^{munu}
& quad – (Lambdaepsilon – LambdaomegaLambda^{-1}a)_mu P^mu
tag{2.4.7}
end{align}
Equating coefficients of $omega_{rhosigma}$ and $epsilon_rho$ on both sides of this equation (and using (2.3.10)) we find
begin{align}
U(Lambda,a) J^{rhosigma} U^{-1}(Lambda,a) & = Lambda_mu^{ rho}Lambda_nu^{ sigma} (J^{munu} – a^mu P^nu + a^nu P^mu)
tag{2.4.8}
U(Lambda,a) P^rho U^{-1}(Lambda,a) & = Lambda_mu^{ rho} P^mu
tag{2.4.9}
end{align}

How are we equating the coefficients? How we find the formula (2.4.8)?

The previous formula that the chapter refers to is

$$ (Lambda^{-1})^{rho}_{nu} = Lambda_{nu}^{rho} = eta_{munu}eta_{rhosigma}Lambda_{sigma}^{nu} tag{2.3.10}$$