TITLE: Abstract proof that internal hom presheaf is a sheaf
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: 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.