<p>Double categories have been extended to (co)lax and virtual double categories. We want to show that the first extension still has a general theory of adjunctions, with examples related to homotopy theory, while the second, wider extension has not. Lax and colax double categories have a finitary weak composition, with associativity comparisons which are not assumed to be invertible. We deal with the <i>colax</i> form (also called oplax), which is related to tensor products of topological ‘algebras’. Double adjunctions can be extended to these structures, in the general ‘colax-lax’ form already studied for (weak) double categories: the left adjoint is colax and the right adjoint is lax. For instance, this is the case of the cylinder-cocylinder adjunction. Now, a normal colax double category is known to be essentially the same as a representable virtual double category. Functors of virtual double categories correspond to <i>lax</i> functors of colax double categories, and can only have adjunctions of the weak-lax form; typically, homotopies will not be represented by a cylinder endofunctor, as we show in a class of examples.</p>