There are two ways a presheaf can fail to be a sheaf.
- It has local sections that should patch together to give a global section, but don't,
- It has non-zero sections which are locally zero.
When dividing the problems into two classes, it is easy to see what sheafifying does. It adds the missing sections from the first problem, and it throws away the extra sections from the second problem.
The latter kind of sections tend to be easier to notice, but are less common. Usually, when a construction or functor must be sheafified, it has local sections that should patch together but don't.
A simple example of a presheaf with this property is the presheaf $F_{p=q}$ of continuous functions on the circle $S^1$ which have the same value at two distinct points $p,q\in S^1$. When I restrict to an open neighborhood of $p$ that doesn't have $q$, the condition on their values goes away. Because the same thing is true for open neighborhoods of $q$ which don't contain $p$, the condition on the functions in this presheaf has no effect on sufficiently small open sets. It follows that this presheaf is locally the same as the sheaf of continuous functions. Therefore, for any function on $S^1$ which has different values on $p$ and $q$, I can restrict it to an open cover where each local section is in $F_{p=q}$, but this function is not in $F_{p=q}$. This is why $F_{p=q}$ is not a sheaf.
When we sheafify, we just add in all these sections, to get the full sheaf of continuous functions. This is clear, because any two sheaves which agree locally are the same (though, I mean that the local sections and local restriction maps agree).
This example really does come up in examples. Consider the map $S^1\rightarrow \infty$, where $\infty$ is the topological space which is $S^1$ with $p$ and $q$ identified. If I pull back the sheaf of functions on $\infty$ in the naive way, the resulting presheaf on $S^1$ is $F_{p=q}$. To get a sheaf, we need to sheafify.
After some research, I think it has not been observed until now. However, all of the bricks needed to make the argument are almost ready.
In paper "Monoidal bicategories and Hopf algebroids" Brain Day and Ross Street defined a notion of convolution in the context of Gray monoids. For a reason that shall become clear later, I am willing to call it "virtual convolution". Here is the definition. Let $\langle A, \delta \colon A \rightarrow A \otimes A, \epsilon \colon A \rightarrow I \rangle$ be a weak comonoid, and $\langle B, \mu \colon B \otimes B \rightarrow B, \eta \colon I \rightarrow B \rangle$ be a weak monoid in a monoidal bi-category with tensor $\otimes$ and unit $I$, then $\langle \hom(A, B), \star, i \rangle$ is a monoidal category by:
\begin{array}{ccc}
f\star g &=& \mu \circ (f \otimes g) \circ \delta \newline
i &=& \eta \circ \epsilon
\end{array}
So the "convolution structure" exists only virtually --- on $\hom$-categories. If the monoidal bi-category admits all right Kan liftings, then such induced monoidal category $\langle \hom(I, B), \star, i \rangle$ for trivial comonoid on $I$ is monoidal (bi)closed by:
$$f \overset{L}\multimap h = \mathit{Rift}_{\mu \circ (f \otimes \mathit{id})}(h)$$
$$f \overset{R}\multimap h = \mathit{Rift}_{\mu \circ (\mathit{id} \otimes f)}(h)$$
Taking for the monoidal bi-category the bi-category of profunctors, we obtain the well-known formula for convolution. However, in the general setting, such induced structure is far weaker than one would wish to have --- for example in the category of profunctors enriched over a monoidal category $\mathbb{V}$ the induced convolution instead of giving a monoidal structure on the category of enriched presheaves:
$$\mathbb{V}^{B^{op}}$$
merely gives a monoidal structure on the underlying category:
$$\hom(I, \mathbb{V}^{B^{op}})$$
Actually, there is a work-around for this issue in the context of enriched categories, as suggested in the paper, but the general weakness of "virtual convolution" is obvious.
The solution is to find a way to "materialize" the convolution. I shall sketch the idea for internal categories. I think all of the following works for split fibrations and split structures, so let me replace the codomain fibration $\mathbb{C}^\rightarrow \rightarrow \mathbb{C}$ from the question by its split version corresponding to the internal "family functor":
$$\mathit{fam}(\mathbb{C}) \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$$
Likewise, for a category $A$ internal to $\mathbb{C}$ I shall write:
$$\mathit{fam}(A) \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$$
for the functor corresponding to the externalisation of $A$. We want to show that given a promonoidal structure $$\langle A, \mu \colon A \times A \nrightarrow A, \eta \colon 1 \nrightarrow A \rangle$$
there is a corresponding monoidal closed structure on:
$$\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}$$
which just means, that each fibre of $\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}$ is a monoidal closed category and reindexing functors preserve these monoidal structures. By fibred Yoneda lemma, for $K \in \mathbb{C}$:
$$\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}(K) = \mathit{Prof}(K, A)$$
where $K$ is interpreted as a discrete internal category. There is a correspondence:
$$\mathit{Prof}(K, A) \approx \mathit{Prof}(1, K^{op} \times A) = \mathit{Prof}(1, K \times A)$$
where the last equality holds because $K^{op} = K$ for any discrete category $K$. Since $K$ has a trivial promonoidal structure:
$$K \times K \overset{\Delta^*}\nrightarrow K$$
we obtain a "product" promonoidal structure on $K \times A$:
\begin{array}{rcc}
K \times A \times K \times A &\overset{\Delta^* \times \mu}\nrightarrow& K \times A \newline
1 &\overset{\langle !^*, \eta \rangle}\nrightarrow& K \times A
\end{array}
In more details, since $\mathbb{C}$ is cartesian, every object $K \in \mathbb{C}$ carries a unique comonoid structure:
\begin{array}{l}
K \overset{\Delta}\rightarrow K \times K \newline
K \overset{!}\rightarrow 1
\end{array}
which has a promonoidal right adjoint structure $\langle \Delta^\*, !^\* \rangle$ in the (bi)category of internal profunctors. The product of the above two promonoidal structures is given by the usual cartesian product of internal categories (note, it is not a product in the bicategory of internal profunctors) followed by the internal product functor $\mathit{fam}(\mathbb{C}) \times \mathit{fam}(\mathbb{C}) \overset{\mathit{prod}}\rightarrow \mathit{fam}(\mathbb{C})$.
Then by "virtual convolution" there is a monoidal (bi)closed structure on $\mathit{Prof}(1, K^{op} \times A)$. Therefore each fibre $\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}(K)$ is a monoidal (bi)closed category. It is easy to check that reindexing functors preserve these structures.
Let me work out the concept of internal Day convolution in case $\mathbb{C} = \mathbf{Set}$ and a promonoidal structure on a small category is monoidal. The split family fibration (or more accurately, the indexed functor corresponding to the family fibration) for a locally small category $A$:
$$\mathit{fam}(A) \colon \mathbf{Set}^{op} \rightarrow \mathbf{Cat}$$
is defined as follows:
\begin{array}{rcl}
\mathit{fam}(A)(K \in \mathbf{Set}) &=& A^K \newline
\mathit{fam}(A)(K \overset{f}\rightarrow L) &=& A^L \overset{(-) \circ f}\rightarrow A^K\newline
\end{array}
where $K, L$ are sets and $K \overset{f}\rightarrow L$ is a function between sets. One may think of category $A^K$ as of the category of $K$-indexed tuples of objects and morphisms from A. Now, given any monoidal structure on a small category $$\langle A, \otimes \colon A \times A \rightarrow A, I \colon 1 \rightarrow A \rangle$$
the usual notion of convolution induces a monoidal structure on $\mathbf{Set}^{A^{op}}$:
$$\langle F, G \rangle \mapsto F \otimes G = \int^{B, C \in A} F(B) \times G(C) \times \hom(-, B \otimes C)$$
The split fibration:
$$\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}} \colon \mathbf{Set}^{op} \rightarrow \mathbf{Cat}$$
may be characterised as follows:
\begin{array}{rcl}
\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}}(K \in \mathbf{Set}) &=& \mathbf{Set}^{A^{op} \times K} \newline
\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}}(K \overset{f}\rightarrow L) &=& \mathbf{Set}^{A^{op} \times L} \overset{(-) \circ (\mathit{id} \times f)}\rightarrow \mathbf{Set}^{A^{op} \times K}\newline
\end{array}
Since $\mathbf{Set}^{A^{op} \times K} \approx (\mathbf{Set}^{A^{op}})^K$ we may think of $\mathbf{Set}^{A^{op} \times K}$ as of $K$-indexed tuples of functors ${A^{op} \rightarrow \mathbf{Set}}$. In fact:
$$\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}} \approx \mathit{fam}(\mathbf{Set}^{A^{op}})$$
It is natural then to extend the monoidal structure induced on $\mathbf{Set}^{A^{op}}$ pointwise to $(\mathbf{Set}^{A^{op}})^K$:
$$(F \otimes G)(k) = \int^{B, C \in A} F(k)(B) \times G(k)(C) \times \hom(-, B \otimes C)$$
where $k \in K$.
On the other hand, using the internal formula for convolution, we get (up to a permutation of arguments):
\begin{array}{c}
\int^{B, C \in A, \beta, \gamma \in K} F(B, \beta) \times G(C, \gamma) \times \hom(\Delta(k), \langle \beta, \gamma \rangle) \times \hom(-, B \otimes C) \newline\hline\newline\hline
\int^{B, C \in A, \beta, \gamma \in K} F(B, \beta) \times G(C, \gamma) \times \hom(k, \beta) \times \hom(k, \gamma) \times \hom(-, B \otimes C) \newline\hline\newline\hline
\int^{B, C \in A} F(B, k) \times G(C, k) \times \hom(-, B \otimes C) \newline
\end{array}
where the first equivalence is the definition of diagonal $\Delta$ --- recall that the diagonal $\Delta(k) = \langle k, k \rangle$ is represented by profunctor $\hom(\langle \overset{1}-, \overset{2}-\rangle, \Delta(\overset{3}-))$, which has profunctorial right adjoint $\hom(\Delta(\overset{1}-), \langle \overset{2}-, \overset{3}-\rangle) \approx \hom(\overset{1}-, \overset{2}-) \times \hom(\overset{1}-, \overset{3}-)$ --- and the second one is by "Yoneda reduction" applied twice.
Final remarks:
Seeing the above proof, one may wonder where the assumptions about the category $\mathbb{C}$ from the question were actually used:
local cartesian closedness guaranteed existence of all right Kan liftings in the bi-category of internal profunctors; without this assumption, the induced monoidal structure on $\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}$ would be generally non-closed; to see that local cartesian closedness is really crucial here, recall that fibration $\mathit{fam}(\mathbb{C})$ is a cartesian closed fibration iff $\mathbb{C}$ is locally cartesian closed ---- this means that without local cartesian closedness even trivial convolution of the monoidal structure on the terminal category is not closed; moreover, which has not been stated in the answer, local cartesian closedness made it possible to speak about internal Yoneda embedding
finite colimits (coequalisers) allowed us to define compositions of internal profunctors
To really obtain a split monoidal closed structure via convolution without moving through the equivalence between Gray monoids and monoidal bi-categories ("Coherence for Tricategories", Gordon, Power, Street), one has (of course!) to replace the monoidal bi-category of internal profunctors by equivalent Gray monoid consisting of internal categories of presheaves and internally cocontinous functors.
I think that the right setting for the concept of Day convolution is a "Yoneda monoidal bi-triangle" as sketched in this answer.
Best Answer
I believe I have the answer in the setting of sheaves of sets.
Let us first do this for presheaves, $Set^{C^{op}}$. This category is Cartesian-closed. This can be seen by setting $Y^X(U):=Hom(X \times U, Y)$, where I have identified $U$ with the representable presheaf $Hom(\cdot,U)$. This is easy to verify. It suffices to show that $Y^X$ agrees with the presheaf $U \mapsto Hom(X|_U,Y|_U)$.
As Peter pointed out, we have the functor $l_U:Set^{C^{op}} \to Set^{C^{op}}/U$ which sends a presheaf $X \mapsto \left(X \times U \to U\right)$, which is right adjoint to the corresponding forgetful functor $Set^{C^{op}}/U \to Set^{C^{op}}$. Here, $X|_U:=l_U(X)$. To explain the notation, note that we have an equivalence of categories $Set^{C^{op}}/U \cong Set^{\left(C/U\right)^{op}}$, so we can think of $l_U$ as restricting $X$ to a presheaf over the slice category $C/U$. Now, given $X$ and $Y$ presheaves on $C$, $Hom(l_U(X),l_U(Y))\cong Hom(X \times U, Y)$ since $l_U$ is a right adjoint to the forgetful functor and the forgetful functor applied to $l_U(X)$ is simply $X \times U$. Hence, we see that $U \mapsto Hom(X|_U,Y|_U)$ agrees with the functor $U \mapsto Hom(X \times U, Y)$.
I claim the same works for a Grothendieck topos:
For this, it suffices to prove that the functor $U \mapsto Hom(X \times U, Y)$ is a sheaf whenever $Y$ is. Let $\left(s_i:U_i \to U\right)_i$ be a cover of the object $U$. Note that $\left(s_i \times id:U_i\times X \to U\times X\right)_i$ is a cover of $U\times X$.
So $\varprojlim \left( \prod \limits_i Y^X(U_i) \rightrightarrows \prod \limits_{i,j} Y^X(U_i\times_U U_j)\right)\cong \varprojlim \left( \prod \limits_i Hom(U_i\times X,Y) \rightrightarrows \prod \limits_{i,j} Hom(U_i\times_U U_j \times X,Y)\right)$
and this is in turn:
$\varprojlim \left( \prod \limits_i Hom(U_i\times X,Y) \rightrightarrows \prod \limits_{i,j} Hom(\left(U_i\times X\right)\times_{U\times X} \left(U_j \times X\right),Y)\right) \cong Hom(U \times X, Y)$
since $Y$ is a sheaf.