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Here are a slightly different flavor of examples. Of course as mentioned by Peter Lumsdain, the trick is always the same: looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"

Consider the category whose objects are set, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).

It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply product, indeed,

A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.

Other similar examples includes:

• The 2-category of sets and span of sets (with the products of sets as monoidal structure).

• The category of small categories and profunctor, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

• The category of polynomial functor (which is actually cartesian closed) described in The cartesian closed bicategory of generalized species and structures by Fiore,Gambino,Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.

But one can somehow "explain the trick" behind them: One way to think about these example is that these categories are equivalent to categories where the Hom objects corresponds more naturally to a natural structure one the set of morphisms, but the equivalence of categories somehow makes the "underlying set" very different.

For example, the category of set and relation can be seen as the category of "Free suplattice":

A suplattice is an ordered set with all supremum, and suplattice morphisms are order preserving map, preserving the supremums. Given two suplattices $X$ and $Y$ the set of morphisms from $X$ and $Y$ is naturally a suplattice for the pointwise ordering (induce by the order of $Y$), and this corresponds to a monoidal symmetric closed structure on the category of suplattice. So this is really a "non-example" of what is asked: the hom objects are really exactly the set of morphisms with a natural structure induced on them.

Now the forgetfull functor from suplattice to sets has a left adjoint, sending any set to the suplattice $P(X)$ of subsets of $X$, sublattice morphisms from $P(X)$ to $P(Y)$ coincide with relation from $X$ and $Y$, and the monoidal structure described above is induced from the one on suplattices...

So our example become a non-example up to an equivalence of categories. Similar description can be obtained for the others example I have mentioned.

Here are a slightly different flavor of examples. Of course as mentioned by Peter Lumsdaine, the trick is always the same: looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"

Consider the category whose objects are sets, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).

It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply the product, indeed,

A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.

Other similar examples include:

• The 2-category of sets and spans of sets (with the products of sets as monoidal structure).

• The category of small categories and profunctors, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

• The category of polynomial functors (which is actually cartesian closed) described in The cartesian closed bicategory of generalized species and structures by Fiore, Gambino, Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.

But one can somehow "explain the trick" behind them: One way to think about these example is that these categories are equivalent to categories where the Hom objects corresponds more naturally to a natural structure on the set of morphisms, but the equivalence of categories somehow makes the "underlying set" very different.

For example, the category of sets and relation can be seen as the category of "Free suplattices":

A suplattice is an ordered set with all supremums, and suplattice morphisms are order preserving map, preserving the supremums. Given two suplattices $X$ and $Y$ the set of morphisms from $X$ and $Y$ is naturally a suplattice for the pointwise ordering (induce by the order of $Y$), and this corresponds to a monoidal symmetric closed structure on the category of suplattice. So this is really a "non-example" of what is asked: the hom objects are really exactly the set of morphisms with a natural structure induced on them.

Now the forgetful functor from suplattice to sets has a left adjoint, sending any set to the suplattice $P(X)$ of subsets of $X$, sublattice morphisms from $P(X)$ to $P(Y)$ coincide with relation from $X$ and $Y$, and the monoidal structure described above is induced from the one on suplattices...

So our example become a non-example up to an equivalence of categories. Similar descriptions can be obtained for the other examples I have mentioned.

Here are a slightly different flavor of examples, of course as mentioned by Peter Lumsdain, the trick is always the same looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"

Consider the category whose objects are set, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).

It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply product, indeed,

A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.

Other similar examples includes:

• The 2-category of sets and span of sets (with the products of sets as monoidal structure).

• The category of small categories and profunctor, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

• The category of polynomial functor (which is actually cartesian closed) described in The cartesian closed bicategory of generalized species and structures by Fiore,Gambino,Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.

But one can somehow "explain the trick" behind these: One way to think about that these example is that these categories are equivalent to categories where the Hom objects corresponds more naturally to a natural structure one the set of morphisms, but the equivalence of categories somehow make the "underlying set" very different.

For example, the category of set and relation can be seen as the category of "Free suplattice":

A suplattice is an ordered set with all supremum, and suplattice morphisms are order preserving map, preserving the supremums. Given two suplattices $X$ and $Y$ the set of morphisms from $X$ and $Y$ is naturally a suplattice for the pointwise ordering (induce by the order of $Y$), and this corresponds to a monoidal symmetric closed structure on the category of suplattice. So this is really a "non-example" of what is asked: the hom objects are really exactly the set of morphisms with a natural structure induced on them.

Now the forgetfull functor from suplattice to sets has a left adjoint, sending any set to the suplattice $P(X)$ of subsets of $X$, sublattice morphisms from $P(X)$ to $P(Y)$ coincide with relation from $X$ and $Y$, and the monoidal structure described above is induced from the one on suplattice...

So our example become a non-example up an equivalence of categories. Similar description can be obtained for the others example I have mentioned.

Here are a slightly different flavor of examples. Of course as mentioned by Peter Lumsdain, the trick is always the same: looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"

Consider the category whose objects are set, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).

It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply product, indeed,

A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.

Other similar examples includes:

• The 2-category of sets and span of sets (with the products of sets as monoidal structure).

• The category of small categories and profunctor, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

• The category of polynomial functor (which is actually cartesian closed) described in The cartesian closed bicategory of generalized species and structures by Fiore,Gambino,Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.

But one can somehow "explain the trick" behind them: One way to think about these example is that these categories are equivalent to categories where the Hom objects corresponds more naturally to a natural structure one the set of morphisms, but the equivalence of categories somehow makes the "underlying set" very different.

For example, the category of set and relation can be seen as the category of "Free suplattice":

A suplattice is an ordered set with all supremum, and suplattice morphisms are order preserving map, preserving the supremums. Given two suplattices $X$ and $Y$ the set of morphisms from $X$ and $Y$ is naturally a suplattice for the pointwise ordering (induce by the order of $Y$), and this corresponds to a monoidal symmetric closed structure on the category of suplattice. So this is really a "non-example" of what is asked: the hom objects are really exactly the set of morphisms with a natural structure induced on them.

Now the forgetfull functor from suplattice to sets has a left adjoint, sending any set to the suplattice $P(X)$ of subsets of $X$, sublattice morphisms from $P(X)$ to $P(Y)$ coincide with relation from $X$ and $Y$, and the monoidal structure described above is induced from the one on suplattices...

So our example become a non-example up to an equivalence of categories. Similar description can be obtained for the others example I have mentioned.

Here are a slightly different flavor of examples, of course as mentioned by Peter Lumsdain, the trick is always the same looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"

Consider the category whose objects are set, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).

It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply product, indeed,

A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.

Other similar examples includes:

• The 2-category of sets and span of sets (with the products of sets as monoidal structure).

• The category of small categories and profunctor, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

• The category of polynomial functor (which is actually cartesian closed) described in The cartesian closed bicategory of generalized species and structures by Fiore,Gambino,Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.

Here are a slightly different flavor of examples, of course as mentioned by Peter Lumsdain, the trick is always the same looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"

Consider the category whose objects are set, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).

It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply product, indeed,

A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.

Other similar examples includes:

• The 2-category of sets and span of sets (with the products of sets as monoidal structure).

• The category of small categories and profunctor, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

• The category of polynomial functor (which is actually cartesian closed) described in The cartesian closed bicategory of generalized species and structures by Fiore,Gambino,Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.

But one can somehow "explain the trick" behind these: One way to think about that these example is that these categories are equivalent to categories where the Hom objects corresponds more naturally to a natural structure one the set of morphisms, but the equivalence of categories somehow make the "underlying set" very different.

For example, the category of set and relation can be seen as the category of "Free suplattice":

A suplattice is an ordered set with all supremum, and suplattice morphisms are order preserving map, preserving the supremums. Given two suplattices $X$ and $Y$ the set of morphisms from $X$ and $Y$ is naturally a suplattice for the pointwise ordering (induce by the order of $Y$), and this corresponds to a monoidal symmetric closed structure on the category of suplattice. So this is really a "non-example" of what is asked: the hom objects are really exactly the set of morphisms with a natural structure induced on them.

Now the forgetfull functor from suplattice to sets has a left adjoint, sending any set to the suplattice $P(X)$ of subsets of $X$, sublattice morphisms from $P(X)$ to $P(Y)$ coincide with relation from $X$ and $Y$, and the monoidal structure described above is induced from the one on suplattice...

So our example become a non-example up an equivalence of categories. Similar description can be obtained for the others example I have mentioned.