  Let (G,X) be a Shimura datum and K a neat open compact subgroup of
$G(\mathbb{A}_f)$. Under mild hypothesis on (G,X), the canonical construction
associates a variation of Hodge structure on $\textrm{Sh}_K(G,X)(\mathbb{C})$
to a representation of G. It is conjectured that this should be of motivic
origin. Specifically, there should be a lift of the canonical construction
which takes values in relative Chow motives over $\textrm{Sh}_K(G,X)$ and is
functorial in (G,X). Using the formalism of mixed Shimura varieties, we show
that such a motivic lift exists on the full subcategory of representations of
Hodge type {(-1,0),(0,-1)}. If (G,X) is equipped with a choice of PEL-datum,
Ancona has defined a motivic lift for all representations of G. We show that
this is independent of the choice of PEL-datum and give criteria for it to be
compatible with base change.
