  The aim of this paper is to construct singular equivalences between functor
categories. As a special case, we show that there exists a singular equivalence
arising from a cotilting module $T$, namely, the singularity category of
$(^\perp T)/[T]$ and that of $(\mod A)/[T]$ are triangle equivalent. In
particular, the canonical module $\omega$ over a commutative Noetherian ring
induces a singular equivalence between $(\mathsf{CM}R)/[\omega]$ and $(\mod
R)/[\omega]$, which generalizes Matsui-Takahashi's theorem. Our result is based
on a sufficient condition for an additive category $\mathcal{A}$ and its
subcategory $\mathcal{X}$ so that the canonical inclusion
$\mathcal{X}\hookrightarrow\mathcal{A}$ induces a singular equivalence
$\mathsf{D_{sg}}(\mathcal{A})\simeq \mathsf{D_{sg}}(\mathcal{X})$, which is a
functor category version of Xiao-Wu Chen's theorem.
