ct.category theory - General construction for internal hom in a presheaf category

The formula for the internal hom between presheaves $Fcolon C^{op}to Set$ and $Gcolon C^{op}to Set$ can be derived from the Yoneda lemma. Given $cin C$, we know that we must have $G^F(c) cong Hom(y(c), G^F) cong Hom(y(c) times F, G)$ so we can simply define $G^F(c) = Hom(y(c) times F, G)$, which is evidently a presheaf on $C$. The isomorphism $Hom(H,G^F)cong Hom(Htimes F, G)$ for non-representable $H$ then follows from the fact that every presheaf $H$ is canonically a colimit of representables, and $Hom(-,G^F)$ and $Hom(-times F,G)$ both preserve colimits (the former by definition of colimits, and the latter by that and since limits and colimits in presheaf categories are computed pointwise and products in $Set$ preserve colimits).

This is Proposition I.6.1 in "Sheaves in geometry and logic."