  In the framework of the Standard Model effective field theory, we examine the
indirect constraints on the trilinear Higgs coupling $\lambda$ that arise from
Higgs production in gluon-gluon-fusion and diphoton Higgs decays. We calculate
2-loop contributions to the $gg \to h$ and $h \to \gamma \gamma$ amplitudes
that are affected by modifications of the trilinear Higgs-boson vertex. This
calculation involves both the computation of anomalous dimensions and finite
matching corrections. Based on our new results, we analyse the sensitivity of
present and future measurements of the $hgg$ and $h \gamma \gamma$ couplings to
shifts in $\lambda$. Under the assumption that $O_6 = - \lambda \left
(H^\dagger H \right )^3$ is the only dimension-6 operator that alters the
trilinear Higgs interactions, we find that at present the considered loop-level
probes provide stronger constraints than $pp \to 2h$. At future high-energy
colliders indirect ${\cal O} (5)$ determinations of the trilinear Higgs
coupling may be possible, making precision measurements of $gg \to h$ and $h
\to \gamma \gamma$ a useful addition to direct extractions of $\lambda$ through
double-Higgs production.
