Weibel gives equivalent conditions for acyclicity of chain complexes, one of which is that the chain complex map $0\to C_.$ is a quasi-isomorphism (i.e. $H_n(0)=0\to H_n(C_.)$ is an isomorphism). While proving the converse, I only used the surjectivity of $H_n(0)=0\to H_n(C_.)$ to get $H_n(C_.)=0$ for all $n$.
Since co-chain complexes are 'dual' to chain complexes, I believe one must reverse all the arrows starting from a chain complex to get a co-chain complex. So, should the condition be like this:
$C^.$ is acyclic $\iff$ $C^.\to 0$ is a quasi-isomorphism
(here one needs injectivity of $H_n(C_.)\to H_n(0)=0$ for the converse).
Or, would it also be fine if condition is like this:
$C^.$ is acyclic $\iff$ $0\to C^.$ is a quasi-isomorphism.
Or does it not matter at all? I am super new to category theory and homological algebra and would like to learn concepts behind what I am writing down as clearly as possible. Any suggestions will be greatly appreciated.