My question is as follows say I have a commutative diagram $\require{AMScd}$ \begin{CD} X @>f>> Y @>g>> Z\\ @V \alpha V V @VV \beta V @VV \gamma V\\ X’ @>>f’> Y @>>g’> Z’ \end{CD}

in a stable $\infty$-category, where the two horizontal rows are fiber/cofiber sequences. Suppose I have another morphism $\delta: X \rightarrow X’$ such that the diagram $\require{AMScd}$ \begin{CD} X @>f>> Y\\ @V \delta V V @VV \beta V\\ X’ @>>f’> Y’ \end{CD}

commutes.

$\textbf{Question:}$ What extra information do I need to conclude that $ \delta \simeq \alpha$ up to some contractile space of homotopies?

Presumably I need something like there should also be a commutative diagram of homotopies: $\require{AMScd}$ \begin{CD} \gamma gf @>\simeq>> g’f’\delta\\ @V \simeq V V @VV \simeq V\\ 0 @>>id> 0 \end{CD}

But I am having a little bit trouble wrapping my head around why this doesn’t “obviously” commute.

Thanks!