  We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to
the theory of motivic fundamental classes of D\'eglise, Jin, and Khan.
  We first compare two kinds of transfers that are naturally defined on
cohomology theories represented by motivic spectra: the framed transfers given
by the recognition principle, which arise from Voevodsky's computation of the
Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin
transfers defined via Verdier's deformation to the normal cone.
  We then introduce the category of finite E-correspondences for E a motivic
ring spectrum, generalizing Voevodsky's category of finite correspondences and
Calm\`es and Fasel's category of finite Milnor-Witt correspondences. Using the
formalism of fundamental classes, we show that the natural functor from the
category of framed correspondences to the category of E-module spectra factors
through the category of finite E-correspondences.
