This paper deals with an extended framework of the distributed asymptotic agreement problem by allowing the presence of unilateral interactions (optimistic or pessimistic) in place of bilateral ones, for a large class of nonlinear monotone time-varying networks. In this original setup we firstly introduce notions of unilateral optimistic and/or pessimistic interaction, of associated bicolored edge in the interaction graph and a suitable graph-theoretical connectedness property. Secondly, we formulate a new assumption of integral connectivity and show that it is sufficient to guarantee exponential convergence towards the agreement subspace. Finally, we show that the proposed conditions are also necessary for consensuability and discuss how the new notions of bicolored graph and related connectivity concepts encompass the usual criteria in the standard case of bilateral interactions. Theoretical advances are emphasized through illustrative examples given both to support the discussion and to highlight how the proposed framework extends all existing conditions for consensus of monotone networks.