The meaning of \((- \otimes v) \dashv (v \multimap -)\) is exactly the condition of \(\multimap\) in a closed symmetric monoidal preorder.
This follows from (1), using the fact that left adjoints preserve joins.
This follows from (1) using the alternative characterization of Galois connections.
Alternatively, start from definition of closed symmetric monoidal preorder and substitute \(v \multimap w\) for \(a\), and note by reflexivity that \((v \multimap w) \leq (v \multimap w)\)
Then we obtain \((v \multimap w) \otimes v \leq w\) (by symmetry of \(\otimes\) we are done)
Since \(v \otimes I = v \leq v\), then the closed condition means that \(v \leq I \multimap v\)
For the other direction, we have \(I \multimap v = I \otimes (I \multimap v) \leq v\) by (3)
We need \(u \otimes (u \multimap v) \otimes (v \multimap w) \leq w\)
This follows from two applications of the third part of this proposition.
Suppose \(\mathcal{V}:=(V,\leq,I,\otimes,\multimap)\) is a closed symmetric monoidal preorder.
For every \(v \in V\) the monotone map \((V, \leq) \xrightarrow{-\otimes v}(V,\leq)\) is left adjoint to \((V, \leq)\ \xrightarrow{v \multimap -} (V,\leq)\)
For any element \(v \in V\) and a subset of elements \(A \subseteq V\), if the join \(\bigvee A\) exists, then so does \(\bigvee_{a \in A} v \otimes a\) and we have \((v \otimes \bigvee A)\cong \bigvee_{a \in A} v \otimes a\)
\(\forall v,w \in V: v \otimes (v \multimap w) \leq w\)
\(\forall v \in V: v \cong (I \multimap v)\)
\(\forall u,v,w \in V: (u \multimap v) \otimes (v \multimap w) \leq (u \multimap w)\)