Does $f \leq f \circ f^\dagger \circ f$ hold in an arbitrary allegory?

Question

If $f$ is an arrow of $\mathrm{Rel}$, then $f \leq f \circ f^\dagger \circ f.$

Proof. Suppose $xy \in f$. Then $xy \in f, yx \in f^\dagger, xy \in f$. Thus $xy \in f \circ f^\dagger \circ f.$

Question. Does this inequality hold in an arbitrary allegory?

Answer 1

Yes, apply the modularity law

$\,uv\,\land w\ \le\ (u\land\, wv^\dagger)v\,$

with $v,w=f:A\to B$ and $u:=1_A$. We get $$f\ =\ 1_Af\land f\ \le\ (1_A\land ff^\dagger)f\ \le\ (ff^\dagger)f\,.$$