# Contractibility of cocartesian liftings

I am searching to show a quite technical result and I am wondering the following. Suppose $$p: C \to D$$ is a functor of infinity categories.

1. Take a cell $$\Delta^2 \to C$$, and suppose that $$\Delta^{\{0,1\}}, \Delta^{\{0,2\}}$$ are p-cocartesian. Is it true that $$\Delta^{\{1,2\}}$$ is p-cocartesian? Some low dimensional examples seems to give a positive answer, but I can't find a general argument not-by-explicit-computation.

2. Suppose $$f:x \to y \in D$$ is an arrow, and fix liftings $$X,Y$$ of the ending points. Is it true that cocartesian liftings of $$f$$ as a subsimplicial set of $$Map(X,Y)$$ are a contractible sSet?

We would have the following corollaries:

1. Suppose $$\sigma: \Delta^2 \to D$$ is a cell and $$j: \Lambda^2_1 \to C$$ is a horn over $$\sigma$$ made of cocartesian edges. Then the infinity category of fillings of $$j$$ over $$\sigma$$ is contractible.

Proof: Indeed, every filling has a third edge which must be cocartesian by (1). Moreover, it lies over a fixed edge in $$\sigma$$, and has fixed ending points. Thus, by (2), the sSet of such fillings is contractible.

1. Take $$H:X \to Y$$ to be a cocartesian lift of $$h: x \to y$$ in $$D$$. Suppose $$h$$ is strictly final in $$D_{x/}$$. Then $$(C_{X/})^{cocar}$$ is weakly contractible.

Proof: Take a $$G:X \to Z$$ in $$(C_{X/})^{cocar}$$ be a cocartesian lift of its image $$g:x \to z$$. By hypothesis, there exist a unique $$f:z \to y$$ from $$g$$ to $$h$$. Thus the set of maps from $$G$$ to $$H$$ lies over a fixed $$f$$, and must be cocartesian by (1). It follows that it is contractible. But then $$H$$ is final in $$(C_{X/})^{cocar}$$, so the latter is weakly contractible.

Thank you! Best, Andrea

1. If $$p$$ is an inner fibration then the answer is yes (HTT.2.4.1.7). Lurie doesn't even define p-cocartesian unless $$p$$ is an inner fibration, but it's not hard to redefine the notion slightly more generally so as to be homotopy invariant (just replace the fiber product in the definition of p-cocartesian with a homotopy fiber product in the Joyal model structure, say).