Given (finite, simple) graphs $G$, $H$ and $K$ and a [homomorphism](https://en.wikipedia.org/wiki/Graph_homomorphism)
$$
G+K\to H+K
$$
where $+$ denotes the [join](http://mathworld.wolfram.com/GraphJoin.html), does it follow that there also exists a graph homomorphism $G\to H$?

If this is known, I'd also appreciate a reference.