The growing availability of single-cell perturbation data called for novel methods to capture treatment response at a single-cell level. While early attempts employed autoencoders, neural optimal transport (OT) emerged as a more principled alternative because it inherently accommodates the challenges of unpaired data induced by cell destruction during data acquisition. However, neural OT relied on casting the problem to convex regression which induced practical challenges during training. The recently introduced Monge Gap overcomes these challenges through a simple and architecturally agnostic regularizer. This approach has proven successful, but it lacks an intrinsic mechanism for generating c-optimal transport maps conditional on covariates present in perturbation response studies (e.g., dosage, time, drug, or cell type). Here, we extend the Monge Gap and propose CMonge, an approach to estimate transport maps conditionally on arbitrary context vectors. It is based on a two-step training procedure combining an encoder-decoder archi-tecture with an OT estimator. We show its value for predicting single-cell perturbation responses, conditional to a drug, a drug dosage, or both. We verify thatour conditional models achieve comparable results to the condition-specific state-of-the-art. Importantly, CMonge learns from data aggregated across conditions which exploits cross-task benefits and allows to generalize to unseen conditionswith promising performance.