On traced monoidal closed categories

The structure theorem of Joyal, Street and Verity says that every traced monoidal category C arises as a monoidal full subcategory of the tortile monoidal category IntC. In this paper we focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into IntC has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts on traced models of linear logic, and some on models of fixed-point computation.