Let \({\mathord {\mathcal{C}}}\) be a \({\mathord {\mathcal{V}}}\)-category and define a \({\mathord {\mathcal{W}}}\)-category \(T_*{\mathord {\mathcal{C}}}\) to have the same objects and to have mapping objects \(T_*{\mathord {\mathcal{C}}}(x,y) :=T{\mathord {\mathcal{C}}}(x,y)\). The composition and identity maps are given by the composites

which make use of the inverses of the natural maps that arise when a finite-product-preserving functor is applied to a finite product. A straightforward diagram chase verifies that \(T_*{\mathord {\mathcal{C}}}\) is a \({\mathord {\mathcal{W}}}\)-category.

If \(F \colon {\mathord {\mathcal{C}}}\to {\mathord {\mathcal{D}}}\) is a \({\mathord {\mathcal{V}}}\)-functor, then we define a \({\mathord {\mathcal{W}}}\)-functor \(T_*F \colon T_*{\mathord {\mathcal{C}}}\to T_*{\mathord {\mathcal{D}}}\) to act on objects by \(c \in {\mathord {\mathcal{C}}}\mapsto Fc \in {\mathord {\mathcal{D}}}\) and with internal action on arrows defined by

Again, a straightforward diagram chase verifies that \(T_*F\) is \({\mathord {\mathcal{W}}}\)-functorial. It is evident from this definition that \(T_*(GF) = T_*G \cdot T_*F\).

Finally, let \(\alpha \colon F \Rightarrow G\) be a \({\mathord {\mathcal{V}}}\)-natural transformation between \({\mathord {\mathcal{V}}}\)-functors \(F,G \colon {\mathord {\mathcal{C}}}\to {\mathord {\mathcal{D}}}\) and define a \({\mathord {\mathcal{W}}}\)-natural transformation \(T_*\alpha \colon T_* F \Rightarrow T_*G\) to have components

Another straightforward diagram chase verifies that \(T_*\alpha \) is \({\mathord {\mathcal{W}}}\)-natural.

It remains to verify this assignment is functorial for both horizontal and vertical composition of enriched natural transformations. The component of \(T_*(\beta \cdot \alpha )\) is defined by the top-horizontal composite below while the component of the vertical composite of \(T_*\alpha \) with \(T_*\beta \colon T_*G \Rightarrow T_*H\) is defined by the bottom composite:

The square commutates by the naturality of the isomorphism \(T(u \times v) \cong Tu \times Tv\), while the triangle commutes because 1 is terminal, so the inverses of the displayed isomorphisms form a commutative triangle. The argument for functoriality of horizontal composites is similar.

Of course right adjoints always preserve products, so the adjoint pair of functors \(F \dashv U\) defines an adjunction in the 2-category of cartesian closed categories and finite-product-preserving functors described in Remark 1.6.5. The 2-functor \({\mathord {\mathcal{V}}}\mapsto {{\mathord {\mathcal{V}}}}\text{-}{\mathord {\mathcal{Cat}}}\) then carries the adjunction displayed on the left to the adjunction displayed on the right.

The internal action \(U_{a,b} \colon U{\mathord {\mathcal{W}}}(a,b) \to {\mathord {\mathcal{V}}}(Ua,Ub)\) of the \({\mathord {\mathcal{V}}}\)-functor \(U \colon U_*{\mathord {\mathcal{W}}}\to {\mathord {\mathcal{V}}}\) is defined by the transpose of the map \(U\textup{ev}\colon U{\mathord {\mathcal{W}}}(a,b) \times Ua \to Ub\) defined by applying \(U\) to the counit of the cartesian closure adjunction of \({\mathord {\mathcal{W}}}\). The \({\mathord {\mathcal{V}}}\)-functoriality of this map follows from naturality of evaluation in a cartesian closed category.

By the \({\mathord {\mathcal{V}}}\)-functoriality of \(U \colon U_*{\mathord {\mathcal{W}}}\to {\mathord {\mathcal{V}}}\), the map

is \({\mathord {\mathcal{V}}}\)-natural in \(w \in U_*{\mathord {\mathcal{W}}}\) for all \(v \in {\mathord {\mathcal{V}}}\). By a general result about enriching adjoints, to construct a compatible \({\mathord {\mathcal{V}}}\)-enrichment of \(F\), we need only demonstrate that this map in an isomorphism in \({\mathord {\mathcal{V}}}\).

We do this by constructing an explicit inverse, namely

where the middle map is defined by applying the unenriched functor \(U\) to the action map from the \({\mathord {\mathcal{W}}}\)-functor \(F \colon F_*{\mathord {\mathcal{V}}}\to {\mathord {\mathcal{W}}}\), which is defined similarly to the \({\mathord {\mathcal{V}}}\)-functor \(U \colon U_*{\mathord {\mathcal{W}}}\to {\mathord {\mathcal{V}}}\).

The proof that these maps are inverses involves a pair of diagram chases, the first of which demonstrates that the top-right composite reduces to the left-bottom composite, which is the identity:

The only subtle point is the commutativity of the trapezoidal region, which expresses the fact that \(\eta \colon \textup{id}_{\mathord {\mathcal{V}}}\Rightarrow UF\) is a closed natural transformation between product-preserving functors between cartesian closed categories. This region commutes because the transposed diagram does:

the right-hand square by naturality, and the left-hand square because any naturally transformation between product-preserving functors is automatically a monoidal natural transformation. The other diagram chase is similar.

Suppose \({\mathord {\mathcal{C}}}\) admits cotensors as a \({\mathord {\mathcal{W}}}\)-category. To verify that \(U_*{\mathord {\mathcal{C}}}\) admits cotensors as a \({\mathord {\mathcal{V}}}\)-category we must supply an isomorphism

in \({\mathord {\mathcal{V}}}\) that is \({\mathord {\mathcal{V}}}\)-natural in \(x\). By the enriched Yoneda lemma, we can extract this isomorphism from an isomorphism

that is \({\mathord {\mathcal{V}}}\)-natural in \(u \in {\mathord {\mathcal{V}}}\). To that end, by composing the \({\mathord {\mathcal{V}}}\)-natural isomorphisms of Lemma 1.6.8, the enriched natural isomorphisms arising from the cartesian closed structure on \({\mathord {\mathcal{V}}}\) and on \(U_*{\mathord {\mathcal{W}}}\), and the isomorphisms that characterize the cotensor on \({\mathord {\mathcal{C}}}\) and express the fact that \(F\) preserves binary products, we have:

The displayed function defines the component at \(v \in {\mathord {\mathcal{V}}}\) of the unique monoidal natural transformation from the underlying set-functor for \({\mathord {\mathcal{V}}}\) to the composite of \(T\) with the underlying set functor for \({\mathord {\mathcal{W}}}\). By the 2-functoriality of Remark 1.6.5, if it defines a monoidal natural isomorphism, then it induces a 2-natural isomorphism between the underlying category 2-functor \((-)_0 \colon {{\mathord {\mathcal{V}}}}\text{-}{\mathord {\mathcal{Cat}}} \to {\mathord {\mathcal{Cat}}}\) and the composite of the change-of-base 2-functor \(T_* \colon {{\mathord {\mathcal{V}}}}\text{-}{\mathord {\mathcal{Cat}}} \to {{\mathord {\mathcal{W}}}}\text{-}{\mathord {\mathcal{Cat}}}\) with the underlying category 2-functor \((-)_0 \colon {{\mathord {\mathcal{W}}}}\text{-}{\mathord {\mathcal{Cat}}} \to {\mathord {\mathcal{Cat}}}\).

Conversely, this condition is necessary for the underlying category of the \({\mathord {\mathcal{W}}}\)-category \(T_*{\mathord {\mathcal{V}}}\) to coincide with the underlying category of the cartesian closed category \({\mathord {\mathcal{V}}}\).

Let \({\mathord {\mathcal{C}}}\) be a \({\mathord {\mathcal{W}}}\) category. Then the hom-set in the underlying category of \(U_*{\mathord {\mathcal{C}}}\) from \(x\) to \(y\) is isomorphic to the corresponding hom-set

in the underlying category of \({\mathord {\mathcal{C}}}\) and moreover this isomorphism respects the composition and identities in the underlying categories. Thus \({\mathord {\mathcal{C}}}_0 \cong U{\mathord {\mathcal{C}}}_0\). A similar argument shows that change of base along \(U\) respects underlying functors and natural transformations.