A certain functor in Hakim's "Topos annellés et schemas relatifs" is a sheaf for the canonical topology In M. Hakim's Topos annellés et schemas relatifs, page 43, (3.4.7), the Author wants to define a sheaf $f_{0A}^*(X)$ over a topos $T$, with respect to the canonical topology. A scheme $X$ is given, together with a commutative ring object $A\in T$, which we can also understand as a representable sheaf over $T$, thus as a functor from $T$ to $Rings$. The definition of $f^*_{0A}$ is as follows: 

$U\in T \mapsto Hom_{Rings}(\Gamma(X,O_X),A(U))$

The definition makes perfectly sense, but it is unclear to my why this gives us a sheaf for the canonical topology. Certainly we should use that $A$ is a (representable) sheaf, but how?
$X$ is fixed, so we could replace $\Gamma(X,O_X)$ with any ring $R$ if this is confusing.
Thank you in advance for any help or suggestion.

EDIT. Perhaps this is simpler than I thought. $A$ is a sheaf by assumption. Hence $Hom_{rings}(R,A(-))$ is a sheaf for any $R$, since $Hom_{Rings}$ is covariant and the equalizer condition for $A$ $$A(U)\to \prod A(U_i)\rightrightarrows \prod A(U_i\times_UU_j)$$ is preserved by applying $Hom_{Rings}(R,-)$ (correct? I am not quite sure of this last sentence).


The argument provided in your edit is correct: The functor $Hom_{\text{Rings}}(R, -)$ preserves limits (this is an entirely general fact, valid for homming out of any object in any category), hence in particular equalizer diagrams.
