Domenico Fiorenza straight

Straightforawrd exercise.

Claim. Ogni freccia $f\colon A\to B$ fattorizza come

per due FS forti innestati $(L_1, R_1)\preceq (L_2, R_2)$.

Proof. Factor $f$ as a composition

with $\lambda_i, l_i\in L_i, \rho_j,r_j\in R_j$.

Now solving the lifting problems

we obtain arrows $s,t,a_1,a_2$. Exploiting the following

Lemma. In a OFS $(L,R)$ the left class is right-3-for-2-closed, namely

and the right class is left-3-for-2-closed, namely

We obtain that $s,t\in L_2\cap R_1$, and that $a_i\in L_i\cap R_i= Iso$. $\blacksquare$