Monoids related to self homotopy equivalences of fibred product

We introduce a nested sequence of monoids related to self-homotopy equivalences of fibrewise pointed spaces, such that the limit is the group of homotopy classes of fibrewise pointed self-equivalences. We explore this monoid for the fibred product in terms of individual spaces. Further we study two related invariants associated to these monoids: self closeness number and self length.