Contrastive Learning has shown impressive results on natural and medical images, without requiring annotated data. However, a particularity of medical images is the availability of meta-data (such as age or sex) that can be exploited for learning representations. Here, we show that the recently proposed contrastive y-Aware InfoNCE loss, that integrates multi-dimensional meta-data, asymptotically optimizes two properties: conditional alignment and global uniformity. Similarly to [33], conditional alignment means that similar samples should have similar features, but conditionally on the meta-data. Instead, global uniformity means that the (normalized) features should be uniformly distributed on the unit hypersphere, independently of the meta-data. Here, we propose to define conditional uniformity, relying on the meta-data, that repel only samples with dissimilar metadata. We show that direct optimization of both conditional alignment and uniformity improves the representations, in terms of linear evaluation, on both CIFAR-100 and a brain MRI dataset.