  In this paper we present the complete one-loop matching conditions, up to
dimension-six operators of the Standard Model effective field theory, resulting
by integrating out the two scalar leptoquarks $S_{1}$ and $S_{3}$. This allows
a phenomenological study of low-energy constraints on this model at one-loop
accuracy, which will be the focus of a subsequent work. Furthermore, it
provides a rich comparison for functional and computational methods for
one-loop matching, that are being developed. As a corollary result, we derive a
complete set of dimension-six operators independent under integration by parts,
but not under equations of motions, called Green's basis, as well as the
complete reduction formulae from this set to the Warsaw basis.
