I've been looking at various general strategies for proving that some category is triangulated, and Lurie manages to prove that a huge class of interesting examples of categories that we know about are triangulated in his book *Higher Algebra* (formerly DAG I-IV and VI).(EDIT: here's a link to the book) The trouble is that I am *very* new to this language, and so what he calls $\infty$-categorical notions that are basic and easily motivated' I see as foreign and unfamiliar.

The part I'm really interested in is the proof of the octahedral axiom on page 24 of *Higher Algebra*. He builds a diagram using a proposition from *Higher Topos Theory* that seems completely out of context (to me!). The proposition says:

``Suppose we are given a diagram of $\infty$-categories $\mathcal{C} \rightarrow \mathcal{D}' \leftarrow \mathcal{D}:p$, where $p$ is a categorical fibration. Let $\mathcal{C}^0$ be a full subcategory of $\mathcal{C}$. Let $\mathcal{K} \subset Map_{\mathcal{D}'}(\mathcal{C}, \mathcal{D})$ be the full subcategory spanned by those functors $F: \mathcal{C} \rightarrow \mathcal{D}$ which are $p$-left Kan extensions of $F\vert\mathcal{C}^0$. Let $\mathcal{K}'\subset \text{Map}_{\mathcal{D}'}(\mathcal{C}^0, \mathcal{D})$ be the full subcategory spanned by those functors $F_0: \mathcal{C}^0 \rightarrow \mathcal{D}$ with the property that, for each object $C \in \mathcal{C}$, the induced diagram $\mathcal{C}^0_{/C} \rightarrow \mathcal{D}$ has a $p$-colimit. Then the restriction functor $\mathcal{K} \rightarrow \mathcal{K}'$ is a trivial fibration of simplicial sets.''

And Lurie says that, in order to prove (TR4), we use this ``repeatedly to construct a map from the nerve of the appropriate partially ordered set into $\mathcal{C}$.'' (See Lurie's book available for download on his webpage.)

Now, obviously this must be some sort of standard use of the proposition, but I would very much like to understand this one proof without reading all of *Higher Topos Theory*, so we have my question:

- Is there possibly a more easy-going reference for this proof? or
- If it doesn't require too much effort, would someone be willing to explain how the cited proposition applies in this instance? or
- Do I really just have to read
*Higher Topos Theory*up through Chapter 4?