Well-defined functor of triangulated category

Question

Let $F \colon D \to D'$ be a morphism of triangulated categories, $S$ a multiplicative system compatible with triangulated structure of $D$.

Supposing the colimits below exist, Lemma 13.14.3 defines a morphism $$ RF(X) = \mathop{\mathrm{colim}}\limits_{s \colon X \to X'} F(X') \longrightarrow RF(Y) = \mathop{\mathrm{colim}}_{s' \colon Y \to Y'} F(Y') $$ where we index over morphisms of $S$ out of $X,Y$ respectively.

The way this is defined uses MS2 of a multiplicative system. It claims that independence follows from MS3 of a multiplicative system.

I don't see exactly why this is true. May someone elaborate?

Answer 1

This is my attempt.

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Take another choice via MS2 $$X' \rightarrow Y'', Y \rightarrow Y''$$ Then we can apply MS2 again to $(Y \rightarrow Y', Y\rightarrow Y'')$ given some $Z$. Then we obersve $$X \rightarrow X ' \rightarrow Y' \rightarrow Z$$ is same as $$X \rightarrow X' \rightarrow Y'' \rightarrow Z$$

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Now apply MS3 to the two morphisms $X ' \rightarrow Z$ which passes thorugh $Y', Y''$ respectively. We construct $$Z \rightarrow Z'$$ a morphism in $S$. Now observe that we have constructed a comutative diagram in $S$ involving $(Y',Y'', Z, Z')$, this commutativity implies that such that both possible maps $FX' \rightarrow RFY$ conincide by definition of $RFY$.