I have a very straightfoward question, why if:
$$gh^{-1} \in \mathbf{K} \implies g \in h\mathbf{K}$$

Where $\mathbf{K}$ is the kernel of a homomorphism.

Also I am aware that the cosets $h\mathbf{K}$ and $\mathbf{K}h$ are the same because $\mathbf{K}$ s a normal subgroup.