
Lemma 13.4.2. Let $\mathcal{D}$ be a pre-triangulated category. For any object $W$ of $\mathcal{D}$ the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, -)$ is homological, and the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, W)$ is cohomological.

Proof. Consider a distinguished triangle $(X, Y, Z, f, g, h)$. We have already seen that $g \circ f = 0$, see Lemma 13.4.1. Suppose $a : W \to Y$ is a morphism such that $g \circ a = 0$. Then we get a commutative diagram

$\xymatrix{ W \ar[r]_1 \ar@{..>}[d]^ b & W \ar[r] \ar[d]^ a & 0 \ar[r] \ar[d]^0 & W[1] \ar@{..>}[d]^{b[1]} \\ X \ar[r] & Y \ar[r] & Z \ar[r] & X[1] }$

Both rows are distinguished triangles (use TR1 for the top row). Hence we can fill the dotted arrow $b$ (first rotate using TR2, then apply TR3, and then rotate back). This proves the lemma. $\square$

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