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We introduce quotient maps in the category of operator systems and show thatthe maximal tensor product is projective with respect to them. Whereas, themaximal tensor product is not injective, which makes the $({\rmel},\max)-nuclearity distinguish a class in the category of operator systems.We generalize Lance's characterization of $C^*$-algebras with the WEP byshowing that $({\rm el},\max)$-nuclearity is equivalent to the weak expectationproperty. Applying Werner's unitization to the dual spaces of operator systems,we consider a class of completely positive maps associated with the maximaltensor product and establish the duality between quotient maps and completeorder embeddings.

On maximal tensor products and quotient maps of operator systems