If a problem A has a poly-time reduction to a problem B in co-NP, is A in co-NP as well?

Question

i.e. $A\leq_pB\:\wedge\:B\in\text{co-NP}\rightarrow A\in\text{co-NP}$ ?

I feel like it's the case but I can't think of a straightforward proof.

Clarification: I am talking about polynomial-time Turing reductions.