= Solution
Here the number of errors means the number of affected physical <qubits>: assume that all errors supported on at most $t$ qubits are jointly correctable. It does not mean the cardinality of an arbitrary selected error list. Let $F$ be any <Pauli operator> of weight at most $2t$. Split its support into disjoint sets $A,B$, each of size at most $t$, and write $F=F_AF_B$. Each Pauli factor is Hermitian, so take $E_a=F_A$, $E_b=F_B$ to get $F=E_a^\dagger E_b$, up to an irrelevant scalar phase if different Pauli representatives are used. The <Knill--Laflamme condition> gives
$$
PFP=PE_a^\dagger E_bP=C_{ab}P.
$$
This is precisely <quantum error detection>: a <projective measurement> with outcomes $P,I-P$ either detects departure from the code or leaves the accepted logical state unchanged up to a scalar. Some errors can be harmless scalars on the code, particularly for degenerate correction; detection is not required to flag such an error with certainty.
Any <linear operator> supported on at most $2t$ qubits is a linear combination of <Pauli operators> on those qubits, so its code compression is scalar by <linearity>. Thus
$$
\boxed{\text{correction of all errors of weight at most }t\ \Longrightarrow\ \text{detection of all errors of weight at most }2t.}
$$
This is <correction implies detection at twice the weight>. The premise about all locations and all error types is essential: merely correcting one selected operator gives no universal statement about every operator of twice its weight.
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