This appears on p. 192 of Land's Introduction to Infinity-Categories which is the first page of his section on Straightening-Unstraightening.
Let $p:\mathscr{E} \to \mathscr{C}$ be a cocartesian fibration between $\infty$-categories. For a map $f:x \to y$ in $\mathscr{C}$, two lifts $z,z' \in \mathscr{E}$ of $x$ and a map $\alpha :z \to z'$ in $\mathscr{E}$ there are $p$-cocartesian lifts $z \to f_!(z)$ and $z' \to f_!(z')$ of $f$ using that $p$ is a cocartesian fibration. Since $z\to f_!(z)$ is $p$-cocartesian we can fill in the bottom arrow (I don't know how to draw dashed arrows on MSE) to obtain a commutative diagram $\require{AMScd}$ \begin{CD} z @>>> z'\\ @V V V @VV V\\ f_!(z) @>>{\exists}> f_!(z') \end{CD} Question. Land then says that there is only a contractible choice of arrows $f_!(z) \to f_!(z')$ making this diagram commutative because $z\to f_!(z)$ is $p$-cocartesian. Why?
I probably missed some result about $p$-cocartesian edges (and/or lack understanding in some result) and would appreciate some clearing up about this.