  This paper introduces a skew variant of the notion of enriched category,
suitable for enrichment over a skew-monoidal category, the main novelty of
which is that the elements of the enriched hom-objects need not be in bijection
with the morphisms of the underlying category. This is the natural setting in
which to introduce the notion of locally weak comonad, which is fundamental to
the theory of enriched algebraic weak factorisation systems. The equivalence,
for a monoidal closed category $\mathcal{V}$, between tensored
$\mathcal{V}$-categories and hommed $\mathcal{V}$-actegories is extended to the
skew setting and easily proved by recognising both skew
$\mathcal{V}$-categories and skew $\mathcal{V}$-actegories as equivalent to
special kinds of skew $\mathcal{V}$-proactegory.
