  We prove a universal property for the $(\infty, n)$-category of
correspondences, generalizing and providing a new proof for the case $n = 2$
from [GR17]. We also provide conditions under which a functor out of a higher
category of correspondences of $\mathcal{C}$ can be extended to a higher
category of correspondences of the free cocompletion of $\mathcal{C}$. These
results will be used in the sequels to this paper to construct $(\infty,
n)$-categorical versions of the theories of quasicoherent and ind-coherent
sheaves in derived algebraic geometry.
