category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A cartesian closed functor is a functor $F\colon \mathcal{C}\to \mathcal{D}$ between cartesian closed categories which preserves both products and exponential objects/internal homs (all the structure of cartesian closed categories).
More precisely, if $F\colon C\to D$ preserves products, then the canonical morphisms $F(A\times B) \to F A \times F B$ (for all objects $A,B \mathcal{C}$) are isomorphisms, and we therefore have canonical induced morphism $F[A,B] \to [F A, F B]$ β the adjuncts of the composite $F[A,B] \times F A \xrightarrow{\cong} F([A,B] \times A) \to F B$. $F$ is cartesian closed if these maps $F[A,B] \to [F A, F B]$ are also isomorphisms.
When cartesian closed categories are identified with cartesian monoidal categories that are also closed monoidal, a cartesian closed functor can be identified with a strong monoidal functor which is also strong closed.
(Frobenius reciprocity)
Let $R : \mathcal{C} \to \mathcal{D}$ be a functor between cartesian closed categories with a left adjoint $L$. Then $R$ is cartesian closed precisely if the natural transformation
is an isomorphism.
The above natural transformation is the mate of the exponential comparison natural transformation $R[A,B] \to [R A, R B]$ under the composite adjunctions
and
This is called the Frobenius reciprocity law. It is discussed, for instance, as (Johnstone, lemma 1.5.8). More generally for closed monoidal categories (not necessarily cartesian monoidal) it is discussed in βWirthmΓΌller contextsβ in
Let still $R$ and $L$ be as above.
If $R$ is full and faithful and $L$ preserves binary products, then $R$ is cartesian closed.
For instance (Johnstone, corollary A1.5.9).
For $\mathcal{C}$ a locally cartesian closed category and $f : X_1 \to X_2$ a morphism, the base change/pullback functor between the slice categories
is cartesian closed.
In particular the inverse image functor of an Γ©tale geometric morphism between toposes is cartesian closed and hence constitutes a cartesian WirthmΓΌller context.
The functor $f^*$ has a left adjoint
given by postcomposition with $f$ (the dependent sum along $f$). Therefore by prop. 1 it is sufficient to show that for all $(A \to X_2)$ in $\mathcal{C}_{/X_2}$ and $(B \stackrel{b}{\to} X_1) \in \mathcal{C}_{/X_1}$ that
in $\mathcal{C}$. But this is the pasting law for pullbacks in $\mathcal{C}$, which says that the two consecutive pullbacks on the left of
are isomorphic to the direct pullback along the composite on the right.
cartesian closed category, locally cartesian closed category
cartesian closed functor, locally cartesian closed functor
cartesian closed model category, locally cartesian closed model category
cartesian closed (β,1)-category locally cartesian closed (β,1)-category
For instance section A1.5 of
Also