Pullback of Quotient Group Isomorphism is Subgroup

Theorem
Let $$G, H$$ be groups.

Let $$N \triangleleft G, K \triangleleft H$$.

Let $$G / N \cong H / K$$ such that $$\theta: G / N \to H / K$$ is such an isomorphism.

The pullback $$G \times^\theta H$$ of $$G$$ and $$H$$ via $$\theta$$ is the subset of $$G \times H$$ of elements of the form $$\left({g, h}\right)$$ where $$\theta \left({g N}\right) = h K$$.

The pullback is a subgroup of $$G \times H$$.

Proof
The fact that the pullback is a subgroup of $$G \times H$$ needs to be checked.