  For the category $\mathscr V$ of complex algebraic varieties, the
Grothendieck group of the commutative monoid of the isomorphism classes of
correspondences $X \xleftarrow f M \xrightarrow g Y$ with proper morphism $f$
and smooth morphism $g$ (such a correspondence is called \emph{a proper-smooth
correspondence}) gives rise to an enriched category $\mathscr Corr(\mathscr
V)^+_{pro-sm}$ of proper-smooth correspondences. In this paper we extend the
well-known theories of characteristic classes of singular varieties such as
Baum-Fulton-MacPherson's Riemann-Roch (abbr. BFM-RR) and MacPherson's Chern
class transformation and so on to this enriched category $\mathscr
Corr(\mathscr V)^+_{pro-sm}$. In order to deal with local complete intersection
(abbr. $\ell.c.i.$) morphism instead of smooth morphism, in a similar manner we
consider an enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$
of \emph{proper-$\ell.c.i.$} zigzags and extend BFM-RR to this enriched
category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$. We also consider an
enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ of proper-smooth
correspondences $(X \xleftarrow f M \xrightarrow g Y; E)$ equipped with complex
vector bundle $E$ on $M$ (such a correspondence is called \emph{a cobordism
bicycle of vector bundle}) and we extend BFM-RR to this enriched category
$\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ as well.
