Adjoint Functors between Closed Categories

The adjoint symmetric functors /, /, remain as in (3.4.3). Additional structure comes into play when the monoidal categories X and Y are closed, in the following sense. 

Definition 3.5.1. A symmetric monoidal closed category (briefly, a closed category) is a symmetric monoidal category 

Commutativity of the second diagram can be deduced from that of the first (and vice-versa), or proved independently. 

Proposition 3.5.5. The following functorial diagrams—where A,B,Gg Xq, E,F,C Yo, Hx, Hy stand for Homxoj Homvo respectively, and with maps arising naturally from those defined above—commute  

Lemma 3.5.5.3. The following diagram with preceding notation) commutes  

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Adjoint Functors between Closed Categories corresponding under tt (3.5.1.2) to the composed map... 

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