  The 3D Bondi-Metzner-Sachs (BMS$_3$) algebra that is the asymptotic symmetry
algebra at null infinity of the $1+2$D asymptotically flat space-time is
isomorphic to the $1+1$D Carrollian conformal algebra. Building on this
connection, various preexisting results in the BMS$_3$-invariant field theories
are reconsidered in light of a purely Carrollian perspective in this paper. In
direct analogy to the covariant transformation laws of the Lorentzian tensors,
the flat Carrollian multiplets are defined and their conformal transformation
properties are established. A first-principle derivation of the Ward identities
in a $1+1$D Carrollian conformal field theory (CCFT) is presented. This
derivation introduces the use of the complex contour-integrals (over the
space-variable) that provide a strong analytic handle to CCFT. The temporal
step-function factors appearing in these Ward identities enable the translation
of the operator product expansions (OPEs) into the language of the operator
commutation relations and vice versa, via a contour-integral prescription.
Motivated by the properties of these step-functions, the $i\epsilon$-forms of
the Ward identities and OPEs are proposed that permit for the hassle-free use
of the algebraic properties of the latter. Finally, utilizing the computational
techniques developed, it is shown that the modes of the quantum energy-momentum
tensor operator generate the centrally extended version of the
infinite-dimensional $1+1$D Carrollian conformal algebra.
