  We present a diagrammatic formulation of recently-revived covariant
functional approaches to one-loop matching from an ultraviolet (UV) theory to a
low-energy effective field theory. Various terms following from a covariant
derivative expansion (CDE) are represented by diagrams which, unlike
conventional Feynman diagrams, involve gauge-covariant quantities and are thus
dubbed "covariant diagrams." The use of covariant diagrams helps organize and
simplify one-loop matching calculations, which we illustrate with examples. Of
particular interest is the derivation of UV model-independent universal
results, which reduce matching calculations of specific UV models to
applications of master formulas. We show how such derivation can be done in a
more concise manner than the previous literature, and discuss how additional
structures that are not directly captured by existing universal results,
including mixed heavy-light loops, open covariant derivatives, and mixed
statistics, can be easily accounted for.
