Stalk of tensor product sheaf is tensor product of stalks via adjunctions and abstract nonsense

Question

Let $(X, \mathcal{O}_{X})$ be a ringed-space with sheaves of modules $\mathcal{F}$ and $\mathcal{G}$. I would like to show that for any point $p \in X$, $$ \mathcal{F}_{p} \otimes_{\mathcal{O}_{X, p}} \mathcal{G}_{p} \simeq (\mathcal{F} \otimes_{\mathcal{O}_{X}} \mathcal{G})_{p} $$ using the fact that left adjoints preserve colimits. I have shown an adjunction between the sheaf tensor and the internal hom functor. The problem is, this seems to only give me that the functor, $$ - \otimes_{\mathcal{O}_{X}} \mathcal{F} $$ preserves colimits. Similarly I have the tensor-hom adjunction for modules over a ring, so I would ideally show this for the tensor product presheaf. But this still only gives me that $$ - \otimes_{\mathcal{O}_{X}(U)} \mathcal{F}(U) $$ preserves colimits. But of course if I want to take a directed limit over open sets containing a point $p$, the ring $\mathcal{O}_{X}(U)$ will change as $U$ changes. Similarly I would also need the result for the bifunctor $$ - \otimes_{\mathcal{O}_{X}} - $$ to get the tensor product of both stalks. Is there any way to use this method to show that the tensor product sheaf commutes with taking stalks, or do I just need to do it manually?