Suppose \(\mathcal{V}=(V,\leq,I,\otimes)\) is a symmetric monoidal preorder that has all joins.
Then \(\mathcal{V}\) is closed, i.e. has a \(\multimap\) operation and hence is a quantale, iff \(\otimes\) distributes over joins
i.e. if Eq (2) from P2.87 holds: \((v \otimes \bigvee A)\cong \bigvee_{a \in A} v \otimes a\).
We proved one direction in P2.87
We need to show that \((v \otimes \bigvee A)\cong \bigvee_{a \in A} v \otimes a\) (and all joins existing) implies that there exists a \(\multimap\) operation that satisfies the closed property: \(\forall a,v,w \in V: (a \otimes v) \leq w\) iff \(a \leq (v \multimap w)\).
The adjoint functor theorem for preorders states that monotone maps preserve joins iff they’re left adjoint, so \(V \xrightarrow{-\otimes v} V\) must have a right adjoint g, which, being a Galois connection, will satisfy the property \((a \otimes v) \leq w \iff a \leq g(w)\) (this is the monoidal closed property).
Let’s rename \(g \equiv v \multimap -\). The adjoint functor theorem even gives us a construction for the right adjoint, namely: \(v \multimap w:=\bigvee\{a \in V\ |\ a \otimes v \leq w\}\).