Stalks of isomorphic presheaves

Question

Let $X$ be a topological space with two presheaves $\mathcal{F}, \mathcal{G}$ on it and $\varphi:\mathcal{F}\rightarrow\mathcal{G}$ an isomorphism. It is well-known that the stalks $\mathcal{F}_P\cong \mathcal{G}_P$ for any $P\in X$. To show such stalks are isomorphic, of course one can check directly that the map $<U,f>\mapsto <U,\varphi(U)(f)>$ on germs of sections is an isomorphism. But my question is that is there a more abstract way using universal property of direct limit to show that $\mathcal{F}_P\cong \mathcal{G}_P$?