Any category with a zero object (e.g., pointed sets) has the property.
Any category with a terminal object $1$ such that every object $C$ is either initial or has a point $1 \to C$ has the property. For example, the category of sets or of topological spaces.
Most toposes do not satisfy the property: if $C$ and $D$ are subterminal, then $K = C\times D = C \wedge D$ usually produces a counterexample.