Consider the following commutative diagram in an abelian category:$$\require{AMScd} \begin{CD} @. A @>{f}>> B @>{g}>> C @>>> 0\\ @. @VV{a}V @VV{b}V @VV{c}V \\ 0@>>> A' @>>{f'}> B' @>>{g'}> C' \end{CD}$$
The snake lemma gives a morphism $\delta:\ker c\to \operatorname{coker} a$ making the sequence $$\ker a \to \ker b\to \ker c\to \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c$$ exact. I wonder if the morphism $\delta$ is unique such that the induced sequence above is exact.
The proof of the snake lemma using elements makes me believe that the answer is yes, but I'm not really sure.
(This is basically the same as Uniqueness of the connecting homomorphism in theory of abelian categories but my question adds a condition, so I think that it was worth making a new question. If someone is against this, I can delete my question.)