Let $\mathcal{A}$ and $\mathcal{B}$ be two $2$-categories and $F : \mathcal{A} \to \mathcal{B}$ be a lax $2$-functor. Given $1$-cells $(f_{i})_{0 \leq i \leq n}$ of $\mathcal{A}$ such that the composition $f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}$ makes sense, this data together with the structural $2$-cells of $F$ give many paths of $2$-cells going from $F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0})$ to $F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0})$, for instance $$ F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0}) \Rightarrow F(f_{n} \circ f_{n-1}) \circ F(f_{n-2}) \circ \cdots \circ F(f_{0}) \Rightarrow \cdots $$ $$\Rightarrow F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}) $$ and $$ F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0}) \Rightarrow F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{1} \circ f_{0}) \Rightarrow \cdots $$ $$\Rightarrow F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}) $$ which correspond to what one gets by "parenthesizing on the left" and "parenthesizing on the right" respectively. It seems to seem obvious that it follows from the definition of lax functor that the $C_{n}$ ways to parenthesize the left hand side all give the same $2$-cell $$ F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0}) \Rightarrow F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}) $$ Since I need this property for a text I am writing, I would like to provide a reference. My question is the following:

Where is this result rigorously stated, and where is it rigorously proved? Hopefully, the two references will be the same.