t-projectivity detected by right t-exactness

Observation 6.4.3. Conversely, if \(P \in C_{\geq 0}\) is a connective object, then \(\hom _C(P,-)\colon C \to \Sp \) is always left t-exact: if \(X \in C_{\leq 0}\), then for \(n > 0\) we have \[ \pi _n(\hom _C(P,X)) \cong \pi _0 \Hom _C(P,X[-n]) = 0, \] since \(P \in C_{\geq 0}\) while \(X[-n] \in C_{\leq -1}\). So \(P\) is t-projective if and only if \(\hom _C(P,-)\) is also right t-exact.