  Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm
Tor}_i^{eRe}(Re,G)=0$ for any $G\in{\rm GProj} eRe$ and $i$ sufficiently large.
Necessary and sufficient conditions are given for the Schur functor $S_e$ to
induce a triangle-equivalence $\mathbb{D}_{def}(R)\simeq\mathbb{D}_{def}(eRe)$.
Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we
provide necessary and sufficient conditions for the singular equivalence
$\mathbb{D}_{sg}(R)\simeq\mathbb{D}_{sg}(eRe)$ to restrict to a
triangle-equivalence $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj}
eRe}$. Applying these to the triangular matrix algebra $T=\left(
  \begin{array}{cc}
  A & M \quad
  0 & B
  \end{array}
  \right)$, corresponding results between candidate categories of $T$ and $A$
(resp. $B$) are obtained. As a consequence, we infer Gorensteinness and
CM-freeness of $T$ from those of $A$ (resp. $B$). Some concrete examples are
given to indicate one can realise the Gorenstein defect category of a
triangular matrix algebra as the singularity category of one of its corner
algabras.
