Standard expressions are connected by atomic braid moves

Each elementary arrow has positive Coxeter length: its one-factor expression is standard by definition, whereas the identity has an empty standard expression, so [F2] rules out an elementary arrow being the identity. Concatenating shortest simple-generator words proves Coxeter-length subadditivity. For any contiguous block of a standard expression, subadditivity gives $\mathrm{\ell }(w)\le {\sum }_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}\mathrm{\ell }(v_i)+\mathrm{\ell }(\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k})\le {\sum }_i\mathrm{\ell }(v_i)=\mathrm{\ell }(w)$; hence the block is standard. In particular, prefixes and suffixes are standard. Induct on the Coxeter length of the common product $w$. At length zero both expressions are empty; removing a first factor from a nonempty standard expression strictly reduces product length.

Let the two nonempty standard words be $U,V$ for $w:I\to K$, and let their first arrows be $e:I\to H$ and $f:I\to H^{\mathrm{\prime }}$. If $e=f$, their remaining words are standard expressions for the same morphism $e^{-1}w:H\to K$ of smaller Coxeter length. Induction connects these tails by finitely many atomic braid moves through standard expressions. Prepending $e$ preserves endpoints and standardness, because every tail has product $e^{-1}w$ and length sum $\mathrm{\ell }(w)-\mathrm{\ell }(e)$.

Otherwise $e\mathrm{\ne }f$. Write $e=v_{a,I}$ and $f=v_{b,I}$. The same source and index determine the same elementary arrow by [F1], so $a\mathrm{\ne }b$. Step 1.1 shows that these distinct arrows are length-additive prefixes of $w$, so [F4] gives the finite relative root difference and a common prefix $q=v_{a,b,I}:I\to L$ with $\mathrm{\ell }(w)=\mathrm{\ell }(q)+\mathrm{\ell }(q^{-1}w)$. By [F5], its two standard words are $A=e{A}^{\mathrm{\prime }}$ and $B=f{B}^{\mathrm{\prime }}$. The remainder $z=q^{-1}w:L\to K$ is a groupoid morphism, so choose a standard word $Z$ for it using [F3]. The words $AZ$ and $BZ$ have the required endpoints, product $w$, and length sum $\mathrm{\ell }(q)+\mathrm{\ell }(z)=\mathrm{\ell }(w)$, so both are standard.

The tails of $U$ and $AZ$ after their common first arrow $e$ are standard words for $e^{-1}w:H\to K$ of strictly smaller Coxeter length. Induction connects them, and prepending $e$ gives a standard chain from $U$ to $AZ$. The same argument with $f$ gives a standard chain from $V$ to $BZ$. Replace $A$ by $B$ in $AZ$ using the atomic braid relation of [F5], keeping $Z$ fixed. This preserves endpoints and standardness, since $A$ and $B$ have the same product $q$ and both have length sum $\mathrm{\ell }(q)$. Concatenate the first chain, this one move, and the reversed second chain. The resulting finite chain connects $U$ to $V$ with every intermediate expression standard, completing the induction.