The question is: Given a functor $F : A^{op} \to \mathsf{Set}$, how do we call an object $?(F)$ in $A$ satisfying the universal property
$\hom(?(F),X) \cong \hom(F,\hom(-,X))$
for all $X \in A$? Some people call it a corepresenting object of $F$. The reason is that a representing object of $F$ is some object $!(F)$ satisfying $\hom(X,!(F)) \cong \hom(\hom(-,X),F)$, since the left hand side simplifies to $F(X)$ by the Yoneda Lemma. Remark that every representing object is also a corepresenting object.
If $F$ is a moduli problem in algebraic geometry, then $?(F)$ with some additional assumptions is usually also called a coarse moduli space (whereas $!(F)$ is the fine moduli space). One of the many references is Definition 2.1. (2) in Adrian Langer's "Moduli Spaces Of Sheaves On Higher Dimensional Varieties", as well as Definition 2.2.1 in "The Geometry of Moduli Spaces of Sheaves" by Huybrecht and Lehn. Perhaps someone can add the original reference.
I will try to answer the second question.
Prop 1. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise. Then, the following are equivalent.
- $\text{lan}_if \dashv \text{lan}_fi$.
- $f$ is the $i$-relative left adjoint of $\text{lan}_fi$, i.e. ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, - ).$
- $f = \text{lift}_{\text{lan}_fi}i$ and the lift is absolute.
If $i$ is only fully faithful $1 \Rightarrow 2$, if $i$ is only dense $2 \Rightarrow 1$.
Proof.
$1 \Rightarrow 2$) $${\bf B}(f, -) \stackrel{i \text{ is ff.}}{\cong} {\bf B}((\text{lan}_if) i, -) \stackrel{1}{\cong} {\bf C}(i, \text{lan}_fi).$$
$2 \Rightarrow 1$)
$${\bf B}(\text{lan}_if, -) \stackrel{\text{point.}}{\cong} \text{lan}_i{\bf B}(f, -) \stackrel{2}{\cong} \text{lan}_i{\bf C}(i, \text{lan}_fi) \stackrel{\text{point.}}{\cong} {\bf C}(\text{lan}_ii, \text{lan}_fi) \stackrel{i \text{ is dense}}{\cong} {\bf C}(-, \text{lan}_fi).$$
$3$ is just a rewriting of $2$.
Now we study a very special setting.
Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise, ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete.
In this setting ${\bf C}$ is a reflective subcategory $ V: {\bf C} \leftrightarrows \text{Set}^{{\bf A}^\circ} : L $ of $\text{Set}^{{\bf A}^\circ}$ via the nerve $V = \text{lan}_i(y_{{\bf A}})$ (V is the right adjoint).
Prop 2. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete. Then, the following are equivalent.
- $\text{lan}_if \dashv \text{lan}_fi$.
- V preserves $\text{lan}_fi$.
- ${\bf C}(i,\text{lan}_fi) \cong \text{lan}_f{\bf C}(i,i)$.
Proof.
Recall that $\text{lan}_if$ is pointwise,because ${\bf B}$ is cocomplete.
$1 \Rightarrow 2)$.
Using Prop 1. we know that ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, - )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, -)$.
Thus, $$ {\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, -) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi)$$
Observe also that $${\bf C}(i, \text{lan}_fi) \cong {\bf C}(Ly_A, \text{lan}_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi),$$ putting the last two equation together, one gets: $$\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi) \cong \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi). $$ By Yoneda Lemma the two functors on the right have to coincide.
$2 \Rightarrow 1).$
Using Prop 1. it is enough to prove that ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, - )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, -)$. Now, $${\bf C}(i, \text{lan}_fi) \cong {\bf C}(Ly_A, \text{lan}_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi) \stackrel{2}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, -). $$
$3$ is just a rewriting of $2$.
Cor. 1 (Quite surprising).
Let ${\bf Set}^{A^\circ} \xleftarrow{y} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $y$ is the Yoneda embedding of a small category and ${\bf B}$ is cocomplete. Then $\text{lan}_yf \dashv \text{lan}_fy$.
Proof 1. In Prop 1 the condition 2 is verified and is essentially a rewriting of the Yoneda Lemma. Moreover $\text{lan}_yf$ is pointwise because ${\bf B}$ is cocomplete.
Proof 2. In Prop 2 the condition 2 is trivially verified because $V$ is the identity.
I shall conclude by saying that the adjunction cannot verify too often.
Rem. 4 Let $y: {\bf A} \to {\bf C}$ be a dense and fully faithful functor. Then the following are equivalent.
- For every map $f: {\bf A} \to {\bf B}$, where ${\bf B}$ is cocomplete, the Kan extensions $\text{lan}_yf$ and $\text{lan}_fy$ exist and are adjoint.
- $y$ is the Yoneda embedding.
Proof. One implications is Cor 1. The other implication can be read as follows, if 1 is verified, then for every functor $f: {\bf A} \to {\bf B}$, there is a unique cocontinous extension ($\text{lan}_yf$), this is the characterization of the free cocompletion under colimits, that is the (small) presheaf construction.
Best Answer
For all $Z \in C^\wedge, Y \in D^\wedge$, we have $C^\wedge(f^\wedge Y,Z)=D^\wedge(Y,f_+ Z)$. If we put $Y = D(-,d), Z = C(-,c)$, we get
$(f_+ C(-,c))(d) = C^\wedge(f^\wedge D(-,d),C(-,c)) = C^\wedge(D(f-,d),C(-,c))$
There seems to be no connection between $f_+ C(-,c)$ and $D(-,fc)$ (only when $f$ is an equivalence). For example,
$D^\wedge(D(-,fc),f_+ C(-,c)) = C^\wedge(f^\wedge D(-,fc),C(-,c)) = C^\wedge(D(f-,fc),C(-,c))$
and it is possible to construct an example where there is no natural transformation $D(f-,fc) \to C(-,c)$ at all. For example if $D(fx,fc)$ is nonempty, but $C(x,c)$ is empty. Take $C^{op}=D=Set, f = Hom(-,2), x = 0, c = 1$.