The holy grail of life's emergence is a capacity for self-reproduction revealed in a chemical system . In a popular scenario, this entity is proposed to be a templating biopolymer, for which what replicates is the sequence of covalently strung monomers. In such a system, replication efficacy does not directly depend on sequence information . The situation with mutually catalytic networks is different: self-reproduction capacity varies significantly among assembly compositions, hence attractor dynamics can lead from very poor to excellent self-reproduction capacity. Our analyses show that this transition may occur with relative ease through simple monomers exchange or chemical modification. However, it is not trivially expected that the reached attractors will also be excellent self-reproducing states. Stationary states in present-day life, e.g. in gene regulation networks or metabolic networks, are often referred to as attractors, but such networks begin with and never deviate from excellent reproduction capacity. In contrast, for the prebiotic networks analyzed here, attractor dynamics is found to tug the system from poor to excellent reproduction capacity.