Extending endofunctors on the category of sets and functions to the category of sets and relations requires one to introduce a certain amount of laxness. This in turn requires us to consider bicategories rather than ordinary categories. The subject of lax extensions of Set-based functors is one of the fundamental components of monoidal topology, an active area of research in categorical algebra. The recent theory of linear bicategories, due to Cockett, Koslowski and Seely, is an extension of the usual notion of bicategory to include a second composition in a way analogous to the two connectives of multiplicative linear logic. It turns out that the category of sets and relations has a second composition making it a linear bicategory. The goal of this thesis is first to define the notion of lax extension of Set-based functors to linear bicategories, and then demonstrate crucial properties of our definition.