  The pion contributions to the coupling $C_0$ of pionless EFT are studied via
both non-relativistic and relativistic forms of chiral effective field theory
for nuclear forces. A definite item in the $2N$-reducible component of the box
diagram is shown to be dominant over the $2N$-irreducible (potential) ones due
to the pinching of low-lying nucleon poles, and this anomalous mapping between
pionless and pionfull EFT's occurs right within the non-relativistic regime. A
natural strategy for renormalization of the pionfull theory emerges as a
byproduct through the interactive analysis of the box diagram. Such mapping
perspective may shed some light on the efficient organization of the pionfull
effective field theory for nuclear forces.
