  We continue our study of tensor products in the operator system category. We
define operator system quotients and exactness in this setting and refine the
notion of nuclearity by studying operator systems that preserve various pairs
of tensor products. One of our main goals is to relate these refinements of
nuclearity to the Kirchberg conjecture. In particular, we prove that the
Kirchberg conjecture is equivalent to the statement that every operator system
that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system
quotients are not always equal to the corresponding operator space quotients
and then study exactness of various operator system tensor products for the
operator system quotient. We prove that an operator system is exact for the min
tensor product if and only if it is (min,el)-nuclear. We give many
characterizations of operator systems that are (min,er)-nulcear,
(el,c)-nuclear, (min,el)-nulcear and (el,max)-nuclear. These characterizations
involve operator system analogues of various properties from the theory of
C*-algebras and operator spaces, including the WEP and LLP.
