  Low-energy effective field theories (EFT) encode information about the
physics at high energies--i.e., the high-energy theory (HET). To extract this
information the EFT and the HET have to be matched to each other. At the
one-loop level, general results for the matching of renormalizable operators
have already been obtained in the literature. In the present paper, we take a
step towards a better understanding of renormalizable operator matching at the
two-loop level: Focusing on the diagrammatic method, we discuss in detail the
various contributions to two-loop matching conditions and compare different
approaches to derive them. Moreover, we discuss which observables are best
suited for the derivation of matching conditions. As a concrete application, we
calculate the $\mathcal{O}\left(\alpha_t \alpha_s \right)$ and $\mathcal{O}
\left(\alpha_t^2 \right)$ matching conditions of the scalar four-point
couplings between the Standard Model (SM) and the Two-Higgs-Doublet Model
(THDM) as well as the THDM and the Minimal Supersymmetric Standard Model
(MSSM). We use the derived formulas to improve the prediction of the SM-like
Higgs mass in the MSSM using the THDM as EFT.
