  If a map $f$ has a homotopy retraction, then Doeraene and El Haouari
conjectured that the sectional category and the relative category of $f$ are
the same. In this work we discuss this conjecture for some lower bounds of
these invariants. In particular, when we consider the diagonal map, we obtain
results supporting Iwase-Sakai's conjecture which asserts that the topological
complexity is the monoidal topological complexity.
