Let \(\mathcal{V}\xrightarrow{f}\mathcal{W}\) be a monoidal monotone map. Given a \(\mathcal{V}\) category, called \(\mathcal{C}\), one can construct an associated \(\mathcal{W}\) category, let’s call it \(\mathcal{C}_f\)
Take the same objects: \(Ob(\mathcal{C}_f):=Ob(\mathcal{C})\)
\(\mathcal{C}_f(a,b) := f(\mathcal{C}(a,b))\)
Check this obeys the definition of an enriched category:
Condition on the monoidal unit:
\(I_W \leq f(I_V)\) — from the first condition of a monoidal monotone map.
\(\forall c \in Ob(\mathcal{C}): I_V \leq \mathcal{C}(c,c)\) — first condition of an enriched category, which \(\mathcal{C}\) is
\(\forall c \in Ob(\mathcal{C}):f(I_V) \leq f(\mathcal{C}(c,c))\) — monotone map property
\(\forall c \in Ob(\mathcal{C}):f(I_V) \leq \mathcal{C}_f(c,c)\) — definition of \(\mathcal{C}_f\)
\(\forall c \in Ob(C_f): I_W \leq C_f(c,c)\) — transitivity, using 1 and 4, noting \(Ob(\mathcal{C})=Ob(\mathcal{C}_f)\)
Condition on monoidal product:
\(\mathcal{C}_f(c,d) \otimes_W \mathcal{C}_f(d,e) = f(\mathcal{C}(c,d)) \otimes_W f(\mathcal{C}(d,e))\) — definition of \(\mathcal{C}_f\)
\(f(\mathcal{C}(c,d)) \otimes_W f(\mathcal{C}(d,e)) \leq f(\mathcal{C}(c,d) \otimes_V \mathcal{C}(d,e))\) — second condition of a monoidal monotone map
\(\mathcal{C}(c,d) \otimes_V \mathcal{C}(d,e) \leq \mathcal{C}(c,e)\) — Second condition of an enriched category
\(f(\mathcal{C}(c,d) \otimes_V \mathcal{C}(d,e)) \leq f(\mathcal{C}(c,e)\) — monotone map property
\(f(\mathcal{C}(c,d) \otimes_V \mathcal{C}(d,e)) \leq \mathcal{C}_f(c,e)\) — definition of \(\mathcal{C}_f\)
\(\mathcal{C}_f(c,d) \otimes_W \mathcal{C}_f(d,e) \leq \mathcal{C}_f(c,e)\) — transitivity, 1,2 and 5