  Additive categories play a fundamental role in mathematics and related
disciplines. Given an additive category equipped with a biadditive functor, one
can construct its category of extensions, which encodes important structural
information. We study how functors between categories of extensions relate to
those at the level of the original categories. When the additive categories in
question are $n$-exangulated, this leads to a characterisation of
$n$-exangulated functors.
  Our approach enables us to study $n$-exangulated categories from a
$2$-categorical perspective. We introduce $n$-exangulated natural
transformations and characterise them using categories of extensions. Our
characterisations allow us to establish a $2$-functor between the
$2$-categories of small $n$-exangulated categories and small exact categories.
A similar result with no smallness assumption is also proved.
  We employ our theory to produce various examples of $n$-exangulated functors
and natural transformations. Although the motivation for this article stems
from representation theory and the study of $n$-exangulated categories, our
results are widely applicable: several require only an additive category
equipped with a biadditive functor with no extra assumptions; others can be
applied by endowing an additive category with its split $n$-exangulated
structure.
