Adjoint Functors between Monoidal Categories

For (b), consider the commutative diagram (with D E Ki, and obvious maps)  

We treat symmetric monoidal categories in this section, leaving the additional closed structure to the next. 

Definition 3.4.2. A symmetric monoidal functor / X — Y between symmetric monoidal categories X, Y is a functor f o- Xq —Yq together with two functorial maps 

Example 3.4.4. (a) Let f X be a map of ringed spaces, X (resp. Y) the category of Ox- (resp. Gy-)modules with its standard structure of symmetric monoidal category having its usual meaning, etc. etc.), and / the usual direct and inverse image functors, see (3.1.7). 

For (3.4.2.1), use the map 7 from (3.2.4.2) and the natural composition Oy f Ox — R/ Ox- One can then deduce via adjointness that R/ is syrmnetric monoidal from the fact that L/ is symmetric monoidal when considered as a functor from Y°P to X°P, see (3.2.4). For this property of If, one can check the requisite commutativity in (3.4.2.2) after replacing each object in X by an isomorphic q-flat complex, and recalling that if A is q-flat, then so is f A (see proof of (3.2.3)(ii)) in view of (3.1.3), the checking is thereby reduced to the context of (a) above, where one can use adjointness (see (3.1.9)) to deduce what needs to be known about f after showing directly that is symmetric monoidal  

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Thus, relations among the four operations can be worked with as instances of category-theoretic relations involving adjoint monoidal functors between... 

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