In a number of recent papers, $(k+l)$-graphs have been
constructed from $k$-graphs by inserting new edges in the last
$l$ dimensions. These constructions have been motivated by
$C^*$-algebraic considerations, so they have not been treated
systematically at the level of higher-rank graphs themselves.
Here we introduce $k$-morphs, which provide a systematic
unifying framework for these various constructions. We think of
$k$-morphs as the analogue, at the level of $k$-graphs, of
$C^*$-correspondences between $C^*$-algebras. To make this
analogy explicit, we introduce a category whose objects are
$k$-graphs and whose morphisms are isomorphism classes of
$k$-morphs. We show how to extend the assignment $\Lambda \mapsto
C^*(\Lambda)$ to a functor from this category to the category
whose objects are $C^*$-algebras and whose morphisms are
isomorphism classes of $C^*$-correspondences.