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Let me recall some definitions first. Let $D$ be a small category and let $J$ be a Grothendieck topology on $D$. A presheaf $F$ on $D$ is called a sheaf when for every covering sieve $\psi \in J(d)$ on some $d \in D$ the natural map $$F(d) \to \lim_{c \to d \in \psi} F(c)$$ is an isomorphism. If $F,G$ are presheaves, then their internal hom is the presheaf $G^F$ defined by $G^F(d) = \hom(F \times \hom(-,d),G)$.

It is not so hard to calculate with elements that if $G$ is a sheaf, then $G^F$ is a sheaf. But I would like to see an abstract proof for this, which uses general principles such as "limits commute with limits". Here is how I would like to start: Let $\psi\in J(d)$. Then $$\lim_{c \to d \in \psi} G^F(c) = \lim_{c \to d \in \psi} \int_a \hom(F(a) \times \hom(a,c),G(a)) = \int_a \lim_{\substack{c \to d \in \psi \\ a \to c}} \hom(F(a),G(a))$$ I am pretty sure that the next step considers the pulled back covering sieve $h^* \phi$ where $h : a \to c$ and uses the sheaf property of $G$ for this sieve.

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"Here is how I would like to start:" ... and I would like to finish this proof. – Martin Brandenburg Jun 21 '14 at 13:37

The shortest abstract nonsense proof I know of goes like this:

  1. $\mathbf{Sh}(\mathcal{C}, J)$ is a reflective subcategory of $\mathbf{Psh}(\mathcal{C})$.
  2. The reflector $\mathbf{Psh}(\mathcal{C}) \to \mathbf{Sh}(\mathcal{C}, J)$ preserves finite products (indeed, finite limits).
  3. A reflective subcategory of a cartesian closed category is an exponential ideal if and only if the reflector preserves finite products.

For the last point, see Proposition 4.3.1 in [Sketches of an elephant, Part A].

If that is too abstract, then here is a slightly different approach. By definition, a sheaf is an object in $\mathbf{Psh}(\mathcal{C})$ that is right orthogonal to the class of $J$-covering sieves $\mathfrak{U} \hookrightarrow \mathcal{C}(-, C)$. Let $\mathcal{M}$ be the class of all morphisms right orthogonal to the class of $J$-covering sieves and let $\mathcal{E}$ be the class of all morphisms left orthogonal to $\mathcal{M}$. Of course, an object that is right orthogonal to $\mathcal{E}$ is still the same thing as a sheaf.

Now, $\mathbf{Sh} (\mathcal{C}, J)$ is an exponential ideal if and only if $\mathcal{E}$ is a product ideal, in the sense that if $f : X \to Y$ is in $\mathcal{E}$ and $Z$ is an object in $\mathcal{E}$, then $f \times \mathrm{id}_Z : X \times Z \to Y \times Z$ is also in $\mathcal{E}$. Indeed, this is just a straightforward exercise in using the exponential adjunction. We are only interested in one direction of the implication, namely the one that says $\mathbf{Sh} (\mathcal{C}, J)$ is an exponential ideal; so (using the comparatively easier fact that $\mathbf{Sh} (\mathcal{C}, J)$ is closed under limits) it suffices to verify the following: if $\mathfrak{U} \hookrightarrow \mathcal{C}(-, C)$ is a $J$-covering sieve, then for any $D$ in $\mathcal{C}$, $\mathfrak{U} \times \mathcal{C}(-, D) \hookrightarrow \mathcal{C}(-, C) \times \mathcal{C}(-, D)$ is in $\mathcal{E}$.

In the end though, one has to do some real work, and this is it. If we assume $\mathcal{C}$ has finite products then the claim is an immediate consequence of the pullback axiom of Grothendieck topologies. Otherwise, let $B$ be an object in $\mathcal{C}$, choose morphisms $B \to C$ and $B \to D$ (i.e. an element of $\mathcal{C}(A, C) \times \mathcal{C}(A, D)$), and let $\mathfrak{V}$ be the sieve consisting of those morphisms $A \to B$ such that the composite $A \to B \to C$ is in $\mathfrak{U}$. Again, by the pullback axiom, $\mathfrak{V}$ is a $J$-covering sieve on $B$. Thus, $\mathfrak{U} \times \mathcal{C}(-, D) \hookrightarrow \mathcal{C}(-, C) \times \mathcal{C}(-, D)$ is the colimit (in $[\mathbb{2}, \mathbf{Psh}(\mathcal{C})]$) of a canonical diagram of $J$-covering sieves. But it is a general fact that $\mathcal{E}$ is closed under colimits, so we are done.

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What about studying sheaf on an elementary topos? Try for example the fifth chapter of "Sheaves in Geometry and Logic" (which you surely know), Lemma 1 page 224 is what you are looking for.

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Well it's not . – Martin Brandenburg Jun 21 '14 at 13:37
Why not? Is it because you cannot understand the proof? – Edoardo Lanari Jun 21 '14 at 13:40
You want "general principles", but this is not a well defined mathematical object. I assume you want to read an element-free proof, let's say "categorical". This uses adjunctions and closure operator; now we're left with two possibilities: either you are adopting a narrow-minded view, for which sheaves are presheaves with nice glueing properties and hence you are a priori not considering my answer, or you want somebody to do your homework (metaphorically speaking), so that even Zhen's answer is not good for you. – Edoardo Lanari Jun 21 '14 at 17:02
@MartinBrandenburg, let $\mathcal{E} = \mathrm{Fun}(D^{\mathrm{op}}, \mathrm{Set})$ be the presheaf topos of your small category $D$. Then by Theorem 2 on page 234, the category $\mathrm{Sh}_j\mathcal{E}$ coincides with the category of $J$-sheaves on $D$, so that the second of the two statements of Lemma 1 on page 224 is precisely what you want. Do you understand now? One has to do some translation, but the proof is there. – Adeel Jun 22 '14 at 8:23
So the problem was that you didn't know that fact, not in the absence of connection. – Edoardo Lanari Jun 22 '14 at 9:57

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