Thus, we have shown that $\bar{\bigtriangledown}_a- \bigtriangledown_a$ defines a map of dual vectors at p to tensors of type $(0,2)$ at $p$.
By property (1), this map is linear. Consequently $(\bar{\bigtriangledown}_a- \bigtriangledown_a)$ defines a tensor of type $(1,2)$ at $p$.
How does this map being linear mean that $(\bar{\bigtriangledown}_a- \bigtriangledown_a)$ has a 1 and not 0 vector spaces? I cannot see how this is deduced.
This is from Wald's General Relativity, page 33.