Generator of intersection of ideals in a PID via adjunction?

Question

In a PID we have the formulas $ \left\langle f\right\rangle + \left\langle g \right\rangle = \left\langle \gcd(f,g) \right\rangle $ and $ \left\langle f\right\rangle \cap \left\langle g \right\rangle = \left\langle \operatorname{lcm}(f,g) \right\rangle $. Now intersection is the product in the lattice of ideals ordered by inclusion, while the $\gcd$ is the product in the lattice of elements of the PID ordered by divisibility. If $ \left\langle \cdot \right\rangle$ had left and right adjoints, this formula would pop out for free. Can $ \left\langle \cdot \right\rangle$ be seen as a functor? Does it have adjoints?