The difference in the action of $\bar{\triangledown_a}$ and $\triangledown_a$ on vector fields and all higher rank tensor fields is determined by
$${\triangledown}_a \omega_b=\bar{\triangledown}_a\omega_b-C^c_{ab}\omega_c $$
, the Liebnitz rule and
4) For all $f \in \mathfrak{F}$ and all $t^a \in V_p$
$$t(f) = t^a\triangledown_a f$$
and 4) tells us that
$$(\bar{\bigtriangledown}_a - \bigtriangledown_a)\omega_bt^b=0$$
I am not sure how this last equality holds? I cannot even see how it is related?
This is from Wald's General Relativity, page 33.