Exponentiable objects in a category, valued in a larger, containing category Recall that when dealing with topological spaces one usually likes dealing with a subcategory of $Top$ which is convenient, one facet of which is that it is cartesian closed. However to get to a similar point with smooth manifolds one needs to consider things like diffeological spaces. Not that there is anything wrong with that. But we have a partial solution if we are just looking for exponentiable objects, and willing to consider infinite-dimensional smooth manifolds (usually Frechet manifolds). 
More formally, an object $A$ of category $C$ with binary products is exponentiable if the functor $-\times A\colon C\to C$ has a right adjoint. The classification of which topological spaces are exponentiable is well known, and cartesian closed categories are defined by the fact that every object is exponentiable.
But in the category of (Hausdorff, finite-dimensional) smooth manifolds the only exponentiable objects are the compact manifolds of dimension at most zero. But we can still sensibly talk about smooth mapping spaces between a general compact manifold and an arbitrary manifold, where the mapping space is an object in the category of Frechet manifolds $Frech$, in which the category $Diff$ of finite-dimensional smooth manifolds sits as a full subcategory.
There are clear analogies with, say, finite CW-complexes, where the 'internal' hom is a topological space of a rather more infinite nature. Similarly, we can consider the mapping presheaf $X \mapsto C(X\times A,Y)$ on C.
What I would like to know is if there is a name for this sort of phenomenon, that we have a category $C$, and full embedding $C\hookrightarrow D$ and $D$-valued mapping objects for certain objects of $C$: these are objects of $C$ which are exponentiable as objects of $D$.
It seems to fall into some gap between cartesian closedness and enrichment, but I don't have a way of making that precise.
 A: Fernando is right that it has something to do with Kan extensions. However, it is not about Kan extensions along an inclusion, but more about extension of an inclusion.
I thought about similar issues a few years ago (and recently --- yesterday) , but in a slightly different context --- after the excellent answer by Todd Trimble to my question Completion of a category, I wondered if there was a general 2-categorical setting that could explain such constructions (I was mainly interested in carrying to a 2-categorical setting the highly related concept of Day convolution). 
Now I try to slowly reproduce some of these ideas.
I shall introduce the concept of a Yoneda triangle (perhaps I should call it the right Yoneda triangle, because there are obvious dual concepts).
Let $\mathbb{W}$ be a 2-category. A Yoneda triangle in $\mathbb{W}$ consists of 1-morphisms $y \colon A \rightarrow \overline{A}$, $f \colon A \rightarrow B$, $g \colon B \rightarrow \overline{A}$ together with a 2-morphism $\eta \colon y \rightarrow g \circ f$ which exhibits $g$ as a pointwise left Kan extension of $y$ along $f$ and exhibits $f$ as an absolute left Kan lifting of $y$ along $g$. (BTW: these data are exactly what led Mark Weber to strengthen the definition of a Yoneda structure introduced by Street and Walters).
The idea of a Yoneda triangle is that, we have a morphism $y \colon A \rightarrow \overline{A}$ which plays the role of a "defective identity" and for a given morphism $f \colon A \rightarrow B$ we try to characterise its right adjoint up to the "defective identity" $y$.
Example: [Yoneda triangles in $\mathbf{Cat}$]
If we take $\mathbb{W}$ to be the 2-category $\mathbf{Cat}$ of locally small categories, functors and natural transformations, then the condition that $G$ is a pointwise left Kan extension of $Y$ along $F$ reduces to:
$$G(-) = \int^{A\in\mathbb{A}} \hom(F(A), -) \times Y(A)$$
(where the coend has to be interpreted as the colimit of $Y$ weighted by $\hom(F(=), -)$ in case the category is not tensored over $\mathbf{Set}$). 
And the condition that $F$ is an absolute left Kan lifting of $Y$ along $G$ reduces to:
$$\hom(Y(-), G(=)) \approx \hom(F(-), =)$$
Particularly, if $Y$ is dense, than $G$ is canonically a pointwise Kan extension --- from density we have:
$$G(-) \approx \int^{A\in\mathbb{A}} \hom(Y(A), G(-)) \times Y(A)$$
and using the formula for an absolute lifting: 
$$G(-) \approx \int^{A\in\mathbb{A}} \hom(F(A), -) \times Y(A)$$
Example: [Adjunction as Yoneda triangle]
It is folklore that an adjunction $f \dashv g$ in a 2-category $\mathbb{W}$ may be equally characterised in the following way: $f$ is an absolute left lifting of the identity along $g$. In such a case $g$ is automatically a pointwise left extension of the identity along $f$ and $\mathit{id}, f, g$ together with the unit of the adjunction form a Yoneda triangle.
Example: [Yoneda triangle as a relative adjunction]
There is an old concept of so called "relative adjunction", which is defined in the same way as the Yoneda triangle, but without the requirement that $g$ is a left Kan extension. Note however, that in such a case $g$ need not be uniquely determined by $f$.
Let me move to the more specific example that you asked about.
Example: [Yoneda triangle along Yoneda embedding]
Let $F \colon \mathbb{A} \rightarrow \mathbb{B}$ be a functor between locally small categories (or more generally, a locally small functor). There is also an inclusion $y_\mathbb{A} \colon \mathbb{A} \rightarrow \mathbf{Set}^{\mathbb{A}^{op}}$. One may easily verify that these data may be always extended to the Yoneda triangle with $G(-) = \hom(F(=), -) \colon \mathbb{B} \rightarrow \mathbf{Set}^{\mathbb{A}^{op}}$ --- which reassembles the fact that every functor always has a "distributional" right adjoint. The same is true for internal categories and for categories enriched in a complete and cocomplete symmetric monoidal closed category, and generally (almost by definition) for any 2-category equipped with a Yoneda structure. 
The essence of the above example is that because the Yoneda functor $y_\mathbb{B} \colon \mathbb{B}\rightarrow \mathbf{Set}^{\mathbb{B}^{op}}$ is a full and faithful embedding, functors $F\colon\mathbb{A} \rightarrow \mathbb{B}$ may be thought as of distributors 
$$y_\mathbb{B} \circ F = \hom(=, F(-))$$
Every distributor arisen in this way has a right adjoint distributor $\hom(F(=), -)$ in the bicategory of distributors. The distributor $\hom(F(=), -)$ has actually the type $\mathbb{B} \rightarrow \mathbf{Set}^{\mathbb{A}^{op}}$, which is the only think that may prevent $F$ of having the ordinary (functorial) right adjoint $G \colon \mathbb{B} \rightarrow \mathbb{A}$ --- just recall, that we say that $F$ has a right adjoint, if there exists $G$ such that:
$$y_\mathbb{A} \circ G \approx \hom(F(=), -)$$
which means:
$$\hom(=, G(-)) \approx \hom(F(=), -)$$
Unfortunately, as a non-mathematician I will not help you with your other examples involving highly mathematical and completely non-understandable terms like a topological space or a manifold, so perhaps you have to calculate the other examples yourself :-)
However, I will give you another example that actually led me to the above considerations. One may similarly define the concept of a Yoneda bi-triangle and a Yoneda monoidal bi-triangle.
Example: [2-powers from Yoneda triangle]
The motivating example is to start with a 2-functor $J \colon \mathbb{W} \rightarrow \mathbb{D}$ equipping a 2-category $\mathbb{W}$ with proarrows, and an extension $Y \colon \mathbb{W} \rightarrow \overline{\mathbb{W}}$ embedding "small objects" into "locally small" (or large) objects in $\overline{\mathbb{W}}$. Then to extend these data to the Yoneda triangle, we have to find a functor $P \colon \mathbb{D} \rightarrow \overline{\mathbb{W}}$ representing a proarrow $A \nrightarrow B$ as a morphism $A \rightarrow P(B)$ in $\overline{\mathbb{W}}$, and a natural transformation $\eta \colon Y \rightarrow P\circ J$ playing the role of a familly of Yoneda morphisms $\eta_A \colon A \rightarrow P(A)$.  
The archetypical situation is when we take $\mathbb{W} = \mathbf{cat}$, $\overline{\mathbb{W}} = \mathbf{Cat}$, $\mathbb{D} = \mathbf{Dist}$, where  $\mathbf{cat}$ is the 2-category of small categories, $\mathbf{Cat}$ is the 2-category of locally small categories, and $\mathbf{Dist}$ is the bicategory of distributors between small categories. Then $J \colon \mathbf{cat} \rightarrow \mathbf{Dist}$, $Y \colon \mathbf{cat} \rightarrow \mathbf{Cat}$ are the usual embeddings, $P \colon \mathbf{Dist} \rightarrow \mathbf{Cat}$ is the covarinat 2-power pseudofunctor $\mathbf{Set}^{(-)^{op}}$ defined on distributors via left Kan extensions, and $\eta_\mathbb{A} \colon \mathbb{A} \rightarrow \mathbf{Set}^{\mathbb{A}^{op}}$ is the Yoneda embedding of a small category $\mathbb{A}$.
We know that there are isomorphisms of categories:
$$\hom_{\mathbf{Dist}}(\mathbb{A}, \mathbb{B}) \approx \hom_{\mathbf{Cat}}(\mathbb{A}, \mathbf{Set}^{\mathbb{B}^{op}})$$
where $\mathbb{A}$ and $\mathbb{B}$ are small. Therefore, to show that $P$ is a (bi)pointwise left Kan extension it suffices to show that $Y$ is 2-dense. However, $Y$ is obviously 2-dense, because the the terminal category is a 2-dense subcategory of $\mathbf{Cat}$ and $Y$ is fully faithful.
The point is that in most situations $\mathbb{D}$ is a monoidal (bi)category, where the monoidal structure is inherited from the closed structure on $\mathbb{W}$. Moreover, functors and the natural transformation constituting the Yoneda triangle are (lax)monoidal. This means that monoids in $\mathbb{W}$ are mapped to the (pro)monoids in $\mathbb{D}$ which are mapped to monoids in $\overline{\mathbb{W}}$. If I am not mistaken this observation leads to an abstract characterisation of the concept of the Day convolution (and in a similar manner one may try to define a Dedekind-MacNeille completion of an object).
In our archetypical situation, a category $\mathbb{A} \times \mathbb{B}$ is mapped by $P$ to $\mathbf{Set}^{\mathbb{A}^{op} \times \mathbb{B}^{op}}$ and the missing morphisms making the unit of the triangle lax monoidal:
$$\mathbf{Set}^{\mathbb{A}^{op}} \times \mathbf{Set}^{\mathbb{B}^{op}} \rightarrow \mathbf{Set}^{\mathbb{A}^{op} \times \mathbb{B}^{op}}$$
is given by the convolution of the distributional identity $\mathbb{A} \times \mathbb{B} \nrightarrow \mathbb{A} \times \mathbb{B}$:
$$\langle F, G \rangle \mapsto \int^{A \in \mathbb{A}, B \in \mathbb{B}} F(A) \times G(B) \times \hom(-, A) \times \hom(=, B) = F(-) \times G(=)$$
Now, a promonoidal category $M \colon \mathbb{A} \times \mathbb{A} \nrightarrow \mathbb{A}$ is mapped by $P$ to:
$$H \mapsto \int^{\langle A, B \rangle \in \mathbb{A}\times \mathbb{A}} H(A, B) \times M(-, A, B)$$
and by composing it with the above map:
$$\langle F, G \rangle \mapsto  \int^{\langle A, B \rangle \in \mathbb{A}\times \mathbb{A}} F(A) \times G(B) \times M(-, A, B)$$
we obtain the well-known formula for convolution.
One may also go in the other direction --- starting from the composition $P \circ J$ satisfying monoidal-like laws and try to find a right or left resolution in the category of (right/left) modules over monoid on $P \circ J$. If I am not mistaken, the left resolution (the Eilenberg-Moore object) of $P \circ J$ in our archetypical situation consists of the category of cocomplete categories and cocontinous functors and the right resolution (the Kleisli object) consists of the bicategory of distributors (i.e. the category of free cocomplete categories and cocontinous functors).
(BTW: this in some sense relates the concept of a proarrow equipment with the concept of a Yoneda structure.)
(BTW: perhaps the concept of a 2-topos should be defined as a Yoneda monoidal bi-triangle induced by the embedding of a 2-category of small objects into a category of bigger objects relatively to a category of "relations" in $\mathbb{W}$, which, for some purposes may be defined as the 2-category of discrete fibred spans, and for another purposes may be defined as the 2-category of codiscrete cofibred cospans).
